segmentation tools in mm - mines paristechcmm.ensmp.fr/~beucher/publi/course_ics_print.pdf · a «...
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ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 1
Serge BEUCHERSerge BEUCHER
CMM ENSMPCMM ENSMP
ICS XII 2007ICS XII 2007Saint EtienneSaint Etienne
September 2007September 2007
SEGMENTATION SEGMENTATION TOOLS in TOOLS in
MATHEMATICAL MATHEMATICAL MORPHOLOGYMORPHOLOGY
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 2
PRELIMINARY REMARKSPRELIMINARY REMARKSbullbull There There isis no no generalgeneral definitiondefinition of image of image segmentationsegmentation
bullbull The The morphologicalmorphological segmentation segmentation approachapproach isis a a pragmaticpragmatic oneone
bullbull NeverthelessNevertheless thisthis approachapproach isisproposingproposing a segmentation a segmentation methodologymethodology a a laquolaquo user guideuser guide raquoraquo for the segmentation for the segmentation toolstools
bullbull It It isis important to important to keepkeep in in mindmind the the variousvarious propertiesproperties of of thesethese toolstools to to avoidavoidsomesome trapstraps TheirTheir implementationimplementation must must bebe as as accurateaccurate as possible to as possible to insureinsure a a good good qualityquality of the of the resultsresults
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 3
bullbull The segmentation tool in MM the Watershed transformationThe segmentation tool in MM the Watershed transformation-- Definition descriptionDefinition description-- How it can be builtHow it can be built-- Bias problems falsitiesBias problems falsities
bullbull Hierarchical segmentationHierarchical segmentation- Focus on the waterfall transformation- Pilings another approach
WHAT IS IT ALL ABOUTWHAT IS IT ALL ABOUT
bullbull How to segment How to segment withwith fhefhe watershedwatershed transformtransform-- The initial The initial ideaidea-- WhyWhy itit doesdoes not not workwork wellwell-- MarkerMarker--controlledcontrolled watershedwatershed-- The segmentation The segmentation toolboxtoolbox and and itsits user user handbookhandbook-- OldOld and new and new toolstools
bullbull Future Future developmentsdevelopments
APPLICATIONAPPLICATIONEXAMPLESEXAMPLES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 4
A MACHINEA MACHINE--TOOL THE WATERSHED TOOL THE WATERSHED TRANSFORM (WTS)TRANSFORM (WTS)
bullbull In MM image segmentation In MM image segmentation isis organisedorganised aroundaround a transformation a transformation the the watershedwatershed transformtransform (1979)(1979)
bullbull The introduction of markers The introduction of markers enhancedenhanced dramaticallydramatically the the efficiencyefficiencyof the of the watershedwatershed (1982)(1982)
bullbull This transformation This transformation belongsbelongs to the to the morphologicalmorphological operatorsoperatorslaquolaquo familyfamily raquoraquo and and especiallyespecially to a class of to a class of operatorsoperators namednamed geodesicgeodesictransformationstransformations
bullbull The WTS The WTS cancan bebe realizedrealized on on variousvarious image structures or image structures or representationsrepresentations 3D images 3D images videosvideos graphs etc This graphs etc This capabilitycapability has has ledled to to hierarchicalhierarchical segmentation solutions (1990)segmentation solutions (1990)
bullbullRecentRecent developmentsdevelopments new new hierarchicalhierarchical toolstools new new criteriacriteria (2005(2005--2007)2007)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 5
bull ItItrsquorsquos a flooding processs a flooding processbull Flooding sources are theFlooding sources are the
minima of the functionminima of the function
Two hierarchies are usedTwo hierarchies are usedbullbull flood progression with theflood progression with the
altitude (sequential process)altitude (sequential process)bullbull flood on the plateausflat zonesflood on the plateausflat zones(parallel process)(parallel process)
The result is a partition of the The result is a partition of the image into catchment basins image into catchment basins and watershed lines (dams)and watershed lines (dams)
THE CLASSICAL WATERSHED ALGORITHMTHE CLASSICAL WATERSHED ALGORITHM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 6
The transformation can be The transformation can be expressed on the successive expressed on the successive levels Z of the function flevels Z of the function f
Use of the geodesic SKIZ Use of the geodesic SKIZ transform to simulate the transform to simulate the propagation without merging propagation without merging
ii
W = m (f)W = m (f)0000The catchment basins at level 0 are the minima at this level
W = W = [[ SKIZ (W )SKIZ (W )] ] 44 m (f) m (f) i+1i+1i+1i+1 Z (f)Z (f)i+1i+1
ii
with
m (f) = Z (f) R (Z (f))m (f) = Z (f) R (Z (f))
R is the geodesic reconstruction
i+1i+1 i+1i+1 Z (f)Z (f) iii+1i+1
THE CLASSICAL WATERSHED ALGORITHM (2)THE CLASSICAL WATERSHED ALGORITHM (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 7
bullbull Classical one (SKIZ using rotation thickenings)Classical one (SKIZ using rotation thickenings)
bullbull Hierarchical queues (a priori order defined in the queue)Hierarchical queues (a priori order defined in the queue)
The use of rotating structuring elements in the SKIZ generates a non isotropic flooding on the plateaus
The tokens in the same stack should be processed at the same time
Most of the watershed algorithms are biased
bullbull WatershedsWatersheds basedbased on graphson graphs
WATERSHED ALGORITHMSWATERSHED ALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 8
For various reasons (complexity computation speed lazinessFor various reasons (complexity computation speed lazinesshelliphellip) ) the unbiased watershed transforms are seldom usedthe unbiased watershed transforms are seldom used
Comparison between a true watershed (left) and the result of a ldquoclassicalrdquo algorithm
These biases may have dramatic consequences in hierarchical These biases may have dramatic consequences in hierarchical approaches based on the comparison of adjacent catchment basinsapproaches based on the comparison of adjacent catchment basins
BIASES AND FALSITIES WITH THE WATERSHEDBIASES AND FALSITIES WITH THE WATERSHED
Due to Due to thisthis biasbias the the watershedwatershed transformtransform isis not UNIQUE (not UNIQUE (shouldshouldbebe uniqueuniquehelliphellip) It ) It dependsdepends on the on the algorithmalgorithm usedused to to buildbuild itit
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 9
In any case the results could not be identical (due to the propagation on the flat zones)
The flooding on the plateaus is based on a MODEL (constant speed) It has mainly two advantages it is simple and it has a physical meaning
The watershed line cannot be built by computing the The watershed line cannot be built by computing the trajectory of rain drops streaming on the topographic surface trajectory of rain drops streaming on the topographic surface (run off) (run off) FORGET ITFORGET IT
BIASES AND FALSITIES WITH THE WATERSHED (2)BIASES AND FALSITIES WITH THE WATERSHED (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 10
A watershed line is not a local feature In particular it is noA watershed line is not a local feature In particular it is not t linked to local structures (crest lines ridgeslinked to local structures (crest lines ridgeshelliphellip) The watershed is ) The watershed is not a LOCAL conceptnot a LOCAL conceptOne canOne canrsquorsquot with the only local knowledge of the neighborhood of a t with the only local knowledge of the neighborhood of a point answer the questionpoint answer the question
Does this point belong to the watershed lineDoes this point belong to the watershed line
BIASES AND FALSITIES WITH THE WTS (3)BIASES AND FALSITIES WITH THE WTS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 11
Is flooding always an ascending processIs flooding always an ascending processIn other words when flooding has reached altitude h is it trueIn other words when flooding has reached altitude h is it true that that ALL the points at a lower altitude have been floodedALL the points at a lower altitude have been flooded
The answer is NO CounterThe answer is NO Counter--example the buttonexample the button--holehole
BIASES AND FALSITIES WITH THE WATERSHED (4)BIASES AND FALSITIES WITH THE WATERSHED (4)
((truetrue buttonbutton--holehole structure in french structure in french GresivaudanGresivaudan))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 12
The watershed transform is used for image segmentationThe watershed transform is used for image segmentation
bullbull GreytoneGreytone segmentationsegmentation
bullbull Shape segmentationShape segmentation
Cutting objects into a union of ldquoconvexrdquo sets by means of the watershed of the distance function
The watershed of the gradient corresponds to the contour lines
USE OF WATERSHEDUSE OF WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 13
Morphological gradientMorphological gradient
g(f) = (f g(f) = (f B) B) -- (f (f 0 0 B)B)Other morphological gradients (half-gradients) can also be definedg-(f) = f - (f 0 B)g+(f) = (f B) ndash fThick gradientsggii(f(f) = (f ) = (f BBii) ) -- (f (f 00 BBii))Regularized gradientsRegularized gradients
THE GRADIENT A REMINDERTHE GRADIENT A REMINDER
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 14
The gradient watershed is overThe gradient watershed is over--segmentedsegmented
To avoid this overTo avoid this over--segmentation due segmentation due to numerous sources of flooding one to numerous sources of flooding one can select some of them (the markers) can select some of them (the markers) and perform the watershed transform and perform the watershed transform controlled by these markerscontrolled by these markers
Gradient images are often noisy and contain a large number of minima Each minimum generates a catchment basin in the WTS
THE MARKERTHE MARKER--CONTROLLED WATERSHEDCONTROLLED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 15
Road segmentationRoad segmentation
Original imageOriginal image
gradientgradient
markersmarkersMarkerMarker--controlled watershed of controlled watershed of
the gradientthe gradient
EXAMPLE OF MARKEREXAMPLE OF MARKER--CONTROLLED CONTROLLED WATERSHEDWATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 16
bullbull Level by level floodingLevel by level flooding
bullbull Hierarchical queuesHierarchical queues
W = M marker setW = M marker set00
W = SKIZ (W )W = SKIZ (W )ii ii--11Z (f) Z (f) 4 4 MMii
This algorithm is simpler than the classical one there is no minima detection
A token at level iltj (the current level) may appear In this case it is treated as a token at level j (the i-queue is no longer alive) With marker-controlled
watershed overflow is the rule and not the exception
MARKERMARKER--CONTROLLED WATERSHED CONTROLLED WATERSHED ALGORITHMSALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 2
PRELIMINARY REMARKSPRELIMINARY REMARKSbullbull There There isis no no generalgeneral definitiondefinition of image of image segmentationsegmentation
bullbull The The morphologicalmorphological segmentation segmentation approachapproach isis a a pragmaticpragmatic oneone
bullbull NeverthelessNevertheless thisthis approachapproach isisproposingproposing a segmentation a segmentation methodologymethodology a a laquolaquo user guideuser guide raquoraquo for the segmentation for the segmentation toolstools
bullbull It It isis important to important to keepkeep in in mindmind the the variousvarious propertiesproperties of of thesethese toolstools to to avoidavoidsomesome trapstraps TheirTheir implementationimplementation must must bebe as as accurateaccurate as possible to as possible to insureinsure a a good good qualityquality of the of the resultsresults
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 3
bullbull The segmentation tool in MM the Watershed transformationThe segmentation tool in MM the Watershed transformation-- Definition descriptionDefinition description-- How it can be builtHow it can be built-- Bias problems falsitiesBias problems falsities
bullbull Hierarchical segmentationHierarchical segmentation- Focus on the waterfall transformation- Pilings another approach
WHAT IS IT ALL ABOUTWHAT IS IT ALL ABOUT
bullbull How to segment How to segment withwith fhefhe watershedwatershed transformtransform-- The initial The initial ideaidea-- WhyWhy itit doesdoes not not workwork wellwell-- MarkerMarker--controlledcontrolled watershedwatershed-- The segmentation The segmentation toolboxtoolbox and and itsits user user handbookhandbook-- OldOld and new and new toolstools
bullbull Future Future developmentsdevelopments
APPLICATIONAPPLICATIONEXAMPLESEXAMPLES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 4
A MACHINEA MACHINE--TOOL THE WATERSHED TOOL THE WATERSHED TRANSFORM (WTS)TRANSFORM (WTS)
bullbull In MM image segmentation In MM image segmentation isis organisedorganised aroundaround a transformation a transformation the the watershedwatershed transformtransform (1979)(1979)
bullbull The introduction of markers The introduction of markers enhancedenhanced dramaticallydramatically the the efficiencyefficiencyof the of the watershedwatershed (1982)(1982)
bullbull This transformation This transformation belongsbelongs to the to the morphologicalmorphological operatorsoperatorslaquolaquo familyfamily raquoraquo and and especiallyespecially to a class of to a class of operatorsoperators namednamed geodesicgeodesictransformationstransformations
bullbull The WTS The WTS cancan bebe realizedrealized on on variousvarious image structures or image structures or representationsrepresentations 3D images 3D images videosvideos graphs etc This graphs etc This capabilitycapability has has ledled to to hierarchicalhierarchical segmentation solutions (1990)segmentation solutions (1990)
bullbullRecentRecent developmentsdevelopments new new hierarchicalhierarchical toolstools new new criteriacriteria (2005(2005--2007)2007)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 5
bull ItItrsquorsquos a flooding processs a flooding processbull Flooding sources are theFlooding sources are the
minima of the functionminima of the function
Two hierarchies are usedTwo hierarchies are usedbullbull flood progression with theflood progression with the
altitude (sequential process)altitude (sequential process)bullbull flood on the plateausflat zonesflood on the plateausflat zones(parallel process)(parallel process)
The result is a partition of the The result is a partition of the image into catchment basins image into catchment basins and watershed lines (dams)and watershed lines (dams)
THE CLASSICAL WATERSHED ALGORITHMTHE CLASSICAL WATERSHED ALGORITHM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 6
The transformation can be The transformation can be expressed on the successive expressed on the successive levels Z of the function flevels Z of the function f
Use of the geodesic SKIZ Use of the geodesic SKIZ transform to simulate the transform to simulate the propagation without merging propagation without merging
ii
W = m (f)W = m (f)0000The catchment basins at level 0 are the minima at this level
W = W = [[ SKIZ (W )SKIZ (W )] ] 44 m (f) m (f) i+1i+1i+1i+1 Z (f)Z (f)i+1i+1
ii
with
m (f) = Z (f) R (Z (f))m (f) = Z (f) R (Z (f))
R is the geodesic reconstruction
i+1i+1 i+1i+1 Z (f)Z (f) iii+1i+1
THE CLASSICAL WATERSHED ALGORITHM (2)THE CLASSICAL WATERSHED ALGORITHM (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 7
bullbull Classical one (SKIZ using rotation thickenings)Classical one (SKIZ using rotation thickenings)
bullbull Hierarchical queues (a priori order defined in the queue)Hierarchical queues (a priori order defined in the queue)
The use of rotating structuring elements in the SKIZ generates a non isotropic flooding on the plateaus
The tokens in the same stack should be processed at the same time
Most of the watershed algorithms are biased
bullbull WatershedsWatersheds basedbased on graphson graphs
WATERSHED ALGORITHMSWATERSHED ALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 8
For various reasons (complexity computation speed lazinessFor various reasons (complexity computation speed lazinesshelliphellip) ) the unbiased watershed transforms are seldom usedthe unbiased watershed transforms are seldom used
Comparison between a true watershed (left) and the result of a ldquoclassicalrdquo algorithm
These biases may have dramatic consequences in hierarchical These biases may have dramatic consequences in hierarchical approaches based on the comparison of adjacent catchment basinsapproaches based on the comparison of adjacent catchment basins
BIASES AND FALSITIES WITH THE WATERSHEDBIASES AND FALSITIES WITH THE WATERSHED
Due to Due to thisthis biasbias the the watershedwatershed transformtransform isis not UNIQUE (not UNIQUE (shouldshouldbebe uniqueuniquehelliphellip) It ) It dependsdepends on the on the algorithmalgorithm usedused to to buildbuild itit
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 9
In any case the results could not be identical (due to the propagation on the flat zones)
The flooding on the plateaus is based on a MODEL (constant speed) It has mainly two advantages it is simple and it has a physical meaning
The watershed line cannot be built by computing the The watershed line cannot be built by computing the trajectory of rain drops streaming on the topographic surface trajectory of rain drops streaming on the topographic surface (run off) (run off) FORGET ITFORGET IT
BIASES AND FALSITIES WITH THE WATERSHED (2)BIASES AND FALSITIES WITH THE WATERSHED (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 10
A watershed line is not a local feature In particular it is noA watershed line is not a local feature In particular it is not t linked to local structures (crest lines ridgeslinked to local structures (crest lines ridgeshelliphellip) The watershed is ) The watershed is not a LOCAL conceptnot a LOCAL conceptOne canOne canrsquorsquot with the only local knowledge of the neighborhood of a t with the only local knowledge of the neighborhood of a point answer the questionpoint answer the question
Does this point belong to the watershed lineDoes this point belong to the watershed line
BIASES AND FALSITIES WITH THE WTS (3)BIASES AND FALSITIES WITH THE WTS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 11
Is flooding always an ascending processIs flooding always an ascending processIn other words when flooding has reached altitude h is it trueIn other words when flooding has reached altitude h is it true that that ALL the points at a lower altitude have been floodedALL the points at a lower altitude have been flooded
The answer is NO CounterThe answer is NO Counter--example the buttonexample the button--holehole
BIASES AND FALSITIES WITH THE WATERSHED (4)BIASES AND FALSITIES WITH THE WATERSHED (4)
((truetrue buttonbutton--holehole structure in french structure in french GresivaudanGresivaudan))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 12
The watershed transform is used for image segmentationThe watershed transform is used for image segmentation
bullbull GreytoneGreytone segmentationsegmentation
bullbull Shape segmentationShape segmentation
Cutting objects into a union of ldquoconvexrdquo sets by means of the watershed of the distance function
The watershed of the gradient corresponds to the contour lines
USE OF WATERSHEDUSE OF WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 13
Morphological gradientMorphological gradient
g(f) = (f g(f) = (f B) B) -- (f (f 0 0 B)B)Other morphological gradients (half-gradients) can also be definedg-(f) = f - (f 0 B)g+(f) = (f B) ndash fThick gradientsggii(f(f) = (f ) = (f BBii) ) -- (f (f 00 BBii))Regularized gradientsRegularized gradients
THE GRADIENT A REMINDERTHE GRADIENT A REMINDER
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 14
The gradient watershed is overThe gradient watershed is over--segmentedsegmented
To avoid this overTo avoid this over--segmentation due segmentation due to numerous sources of flooding one to numerous sources of flooding one can select some of them (the markers) can select some of them (the markers) and perform the watershed transform and perform the watershed transform controlled by these markerscontrolled by these markers
Gradient images are often noisy and contain a large number of minima Each minimum generates a catchment basin in the WTS
THE MARKERTHE MARKER--CONTROLLED WATERSHEDCONTROLLED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 15
Road segmentationRoad segmentation
Original imageOriginal image
gradientgradient
markersmarkersMarkerMarker--controlled watershed of controlled watershed of
the gradientthe gradient
EXAMPLE OF MARKEREXAMPLE OF MARKER--CONTROLLED CONTROLLED WATERSHEDWATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 16
bullbull Level by level floodingLevel by level flooding
bullbull Hierarchical queuesHierarchical queues
W = M marker setW = M marker set00
W = SKIZ (W )W = SKIZ (W )ii ii--11Z (f) Z (f) 4 4 MMii
This algorithm is simpler than the classical one there is no minima detection
A token at level iltj (the current level) may appear In this case it is treated as a token at level j (the i-queue is no longer alive) With marker-controlled
watershed overflow is the rule and not the exception
MARKERMARKER--CONTROLLED WATERSHED CONTROLLED WATERSHED ALGORITHMSALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 3
bullbull The segmentation tool in MM the Watershed transformationThe segmentation tool in MM the Watershed transformation-- Definition descriptionDefinition description-- How it can be builtHow it can be built-- Bias problems falsitiesBias problems falsities
bullbull Hierarchical segmentationHierarchical segmentation- Focus on the waterfall transformation- Pilings another approach
WHAT IS IT ALL ABOUTWHAT IS IT ALL ABOUT
bullbull How to segment How to segment withwith fhefhe watershedwatershed transformtransform-- The initial The initial ideaidea-- WhyWhy itit doesdoes not not workwork wellwell-- MarkerMarker--controlledcontrolled watershedwatershed-- The segmentation The segmentation toolboxtoolbox and and itsits user user handbookhandbook-- OldOld and new and new toolstools
bullbull Future Future developmentsdevelopments
APPLICATIONAPPLICATIONEXAMPLESEXAMPLES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 4
A MACHINEA MACHINE--TOOL THE WATERSHED TOOL THE WATERSHED TRANSFORM (WTS)TRANSFORM (WTS)
bullbull In MM image segmentation In MM image segmentation isis organisedorganised aroundaround a transformation a transformation the the watershedwatershed transformtransform (1979)(1979)
bullbull The introduction of markers The introduction of markers enhancedenhanced dramaticallydramatically the the efficiencyefficiencyof the of the watershedwatershed (1982)(1982)
bullbull This transformation This transformation belongsbelongs to the to the morphologicalmorphological operatorsoperatorslaquolaquo familyfamily raquoraquo and and especiallyespecially to a class of to a class of operatorsoperators namednamed geodesicgeodesictransformationstransformations
bullbull The WTS The WTS cancan bebe realizedrealized on on variousvarious image structures or image structures or representationsrepresentations 3D images 3D images videosvideos graphs etc This graphs etc This capabilitycapability has has ledled to to hierarchicalhierarchical segmentation solutions (1990)segmentation solutions (1990)
bullbullRecentRecent developmentsdevelopments new new hierarchicalhierarchical toolstools new new criteriacriteria (2005(2005--2007)2007)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 5
bull ItItrsquorsquos a flooding processs a flooding processbull Flooding sources are theFlooding sources are the
minima of the functionminima of the function
Two hierarchies are usedTwo hierarchies are usedbullbull flood progression with theflood progression with the
altitude (sequential process)altitude (sequential process)bullbull flood on the plateausflat zonesflood on the plateausflat zones(parallel process)(parallel process)
The result is a partition of the The result is a partition of the image into catchment basins image into catchment basins and watershed lines (dams)and watershed lines (dams)
THE CLASSICAL WATERSHED ALGORITHMTHE CLASSICAL WATERSHED ALGORITHM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 6
The transformation can be The transformation can be expressed on the successive expressed on the successive levels Z of the function flevels Z of the function f
Use of the geodesic SKIZ Use of the geodesic SKIZ transform to simulate the transform to simulate the propagation without merging propagation without merging
ii
W = m (f)W = m (f)0000The catchment basins at level 0 are the minima at this level
W = W = [[ SKIZ (W )SKIZ (W )] ] 44 m (f) m (f) i+1i+1i+1i+1 Z (f)Z (f)i+1i+1
ii
with
m (f) = Z (f) R (Z (f))m (f) = Z (f) R (Z (f))
R is the geodesic reconstruction
i+1i+1 i+1i+1 Z (f)Z (f) iii+1i+1
THE CLASSICAL WATERSHED ALGORITHM (2)THE CLASSICAL WATERSHED ALGORITHM (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 7
bullbull Classical one (SKIZ using rotation thickenings)Classical one (SKIZ using rotation thickenings)
bullbull Hierarchical queues (a priori order defined in the queue)Hierarchical queues (a priori order defined in the queue)
The use of rotating structuring elements in the SKIZ generates a non isotropic flooding on the plateaus
The tokens in the same stack should be processed at the same time
Most of the watershed algorithms are biased
bullbull WatershedsWatersheds basedbased on graphson graphs
WATERSHED ALGORITHMSWATERSHED ALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 8
For various reasons (complexity computation speed lazinessFor various reasons (complexity computation speed lazinesshelliphellip) ) the unbiased watershed transforms are seldom usedthe unbiased watershed transforms are seldom used
Comparison between a true watershed (left) and the result of a ldquoclassicalrdquo algorithm
These biases may have dramatic consequences in hierarchical These biases may have dramatic consequences in hierarchical approaches based on the comparison of adjacent catchment basinsapproaches based on the comparison of adjacent catchment basins
BIASES AND FALSITIES WITH THE WATERSHEDBIASES AND FALSITIES WITH THE WATERSHED
Due to Due to thisthis biasbias the the watershedwatershed transformtransform isis not UNIQUE (not UNIQUE (shouldshouldbebe uniqueuniquehelliphellip) It ) It dependsdepends on the on the algorithmalgorithm usedused to to buildbuild itit
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 9
In any case the results could not be identical (due to the propagation on the flat zones)
The flooding on the plateaus is based on a MODEL (constant speed) It has mainly two advantages it is simple and it has a physical meaning
The watershed line cannot be built by computing the The watershed line cannot be built by computing the trajectory of rain drops streaming on the topographic surface trajectory of rain drops streaming on the topographic surface (run off) (run off) FORGET ITFORGET IT
BIASES AND FALSITIES WITH THE WATERSHED (2)BIASES AND FALSITIES WITH THE WATERSHED (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 10
A watershed line is not a local feature In particular it is noA watershed line is not a local feature In particular it is not t linked to local structures (crest lines ridgeslinked to local structures (crest lines ridgeshelliphellip) The watershed is ) The watershed is not a LOCAL conceptnot a LOCAL conceptOne canOne canrsquorsquot with the only local knowledge of the neighborhood of a t with the only local knowledge of the neighborhood of a point answer the questionpoint answer the question
Does this point belong to the watershed lineDoes this point belong to the watershed line
BIASES AND FALSITIES WITH THE WTS (3)BIASES AND FALSITIES WITH THE WTS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 11
Is flooding always an ascending processIs flooding always an ascending processIn other words when flooding has reached altitude h is it trueIn other words when flooding has reached altitude h is it true that that ALL the points at a lower altitude have been floodedALL the points at a lower altitude have been flooded
The answer is NO CounterThe answer is NO Counter--example the buttonexample the button--holehole
BIASES AND FALSITIES WITH THE WATERSHED (4)BIASES AND FALSITIES WITH THE WATERSHED (4)
((truetrue buttonbutton--holehole structure in french structure in french GresivaudanGresivaudan))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 12
The watershed transform is used for image segmentationThe watershed transform is used for image segmentation
bullbull GreytoneGreytone segmentationsegmentation
bullbull Shape segmentationShape segmentation
Cutting objects into a union of ldquoconvexrdquo sets by means of the watershed of the distance function
The watershed of the gradient corresponds to the contour lines
USE OF WATERSHEDUSE OF WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 13
Morphological gradientMorphological gradient
g(f) = (f g(f) = (f B) B) -- (f (f 0 0 B)B)Other morphological gradients (half-gradients) can also be definedg-(f) = f - (f 0 B)g+(f) = (f B) ndash fThick gradientsggii(f(f) = (f ) = (f BBii) ) -- (f (f 00 BBii))Regularized gradientsRegularized gradients
THE GRADIENT A REMINDERTHE GRADIENT A REMINDER
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 14
The gradient watershed is overThe gradient watershed is over--segmentedsegmented
To avoid this overTo avoid this over--segmentation due segmentation due to numerous sources of flooding one to numerous sources of flooding one can select some of them (the markers) can select some of them (the markers) and perform the watershed transform and perform the watershed transform controlled by these markerscontrolled by these markers
Gradient images are often noisy and contain a large number of minima Each minimum generates a catchment basin in the WTS
THE MARKERTHE MARKER--CONTROLLED WATERSHEDCONTROLLED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 15
Road segmentationRoad segmentation
Original imageOriginal image
gradientgradient
markersmarkersMarkerMarker--controlled watershed of controlled watershed of
the gradientthe gradient
EXAMPLE OF MARKEREXAMPLE OF MARKER--CONTROLLED CONTROLLED WATERSHEDWATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 16
bullbull Level by level floodingLevel by level flooding
bullbull Hierarchical queuesHierarchical queues
W = M marker setW = M marker set00
W = SKIZ (W )W = SKIZ (W )ii ii--11Z (f) Z (f) 4 4 MMii
This algorithm is simpler than the classical one there is no minima detection
A token at level iltj (the current level) may appear In this case it is treated as a token at level j (the i-queue is no longer alive) With marker-controlled
watershed overflow is the rule and not the exception
MARKERMARKER--CONTROLLED WATERSHED CONTROLLED WATERSHED ALGORITHMSALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 4
A MACHINEA MACHINE--TOOL THE WATERSHED TOOL THE WATERSHED TRANSFORM (WTS)TRANSFORM (WTS)
bullbull In MM image segmentation In MM image segmentation isis organisedorganised aroundaround a transformation a transformation the the watershedwatershed transformtransform (1979)(1979)
bullbull The introduction of markers The introduction of markers enhancedenhanced dramaticallydramatically the the efficiencyefficiencyof the of the watershedwatershed (1982)(1982)
bullbull This transformation This transformation belongsbelongs to the to the morphologicalmorphological operatorsoperatorslaquolaquo familyfamily raquoraquo and and especiallyespecially to a class of to a class of operatorsoperators namednamed geodesicgeodesictransformationstransformations
bullbull The WTS The WTS cancan bebe realizedrealized on on variousvarious image structures or image structures or representationsrepresentations 3D images 3D images videosvideos graphs etc This graphs etc This capabilitycapability has has ledled to to hierarchicalhierarchical segmentation solutions (1990)segmentation solutions (1990)
bullbullRecentRecent developmentsdevelopments new new hierarchicalhierarchical toolstools new new criteriacriteria (2005(2005--2007)2007)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 5
bull ItItrsquorsquos a flooding processs a flooding processbull Flooding sources are theFlooding sources are the
minima of the functionminima of the function
Two hierarchies are usedTwo hierarchies are usedbullbull flood progression with theflood progression with the
altitude (sequential process)altitude (sequential process)bullbull flood on the plateausflat zonesflood on the plateausflat zones(parallel process)(parallel process)
The result is a partition of the The result is a partition of the image into catchment basins image into catchment basins and watershed lines (dams)and watershed lines (dams)
THE CLASSICAL WATERSHED ALGORITHMTHE CLASSICAL WATERSHED ALGORITHM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 6
The transformation can be The transformation can be expressed on the successive expressed on the successive levels Z of the function flevels Z of the function f
Use of the geodesic SKIZ Use of the geodesic SKIZ transform to simulate the transform to simulate the propagation without merging propagation without merging
ii
W = m (f)W = m (f)0000The catchment basins at level 0 are the minima at this level
W = W = [[ SKIZ (W )SKIZ (W )] ] 44 m (f) m (f) i+1i+1i+1i+1 Z (f)Z (f)i+1i+1
ii
with
m (f) = Z (f) R (Z (f))m (f) = Z (f) R (Z (f))
R is the geodesic reconstruction
i+1i+1 i+1i+1 Z (f)Z (f) iii+1i+1
THE CLASSICAL WATERSHED ALGORITHM (2)THE CLASSICAL WATERSHED ALGORITHM (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 7
bullbull Classical one (SKIZ using rotation thickenings)Classical one (SKIZ using rotation thickenings)
bullbull Hierarchical queues (a priori order defined in the queue)Hierarchical queues (a priori order defined in the queue)
The use of rotating structuring elements in the SKIZ generates a non isotropic flooding on the plateaus
The tokens in the same stack should be processed at the same time
Most of the watershed algorithms are biased
bullbull WatershedsWatersheds basedbased on graphson graphs
WATERSHED ALGORITHMSWATERSHED ALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 8
For various reasons (complexity computation speed lazinessFor various reasons (complexity computation speed lazinesshelliphellip) ) the unbiased watershed transforms are seldom usedthe unbiased watershed transforms are seldom used
Comparison between a true watershed (left) and the result of a ldquoclassicalrdquo algorithm
These biases may have dramatic consequences in hierarchical These biases may have dramatic consequences in hierarchical approaches based on the comparison of adjacent catchment basinsapproaches based on the comparison of adjacent catchment basins
BIASES AND FALSITIES WITH THE WATERSHEDBIASES AND FALSITIES WITH THE WATERSHED
Due to Due to thisthis biasbias the the watershedwatershed transformtransform isis not UNIQUE (not UNIQUE (shouldshouldbebe uniqueuniquehelliphellip) It ) It dependsdepends on the on the algorithmalgorithm usedused to to buildbuild itit
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 9
In any case the results could not be identical (due to the propagation on the flat zones)
The flooding on the plateaus is based on a MODEL (constant speed) It has mainly two advantages it is simple and it has a physical meaning
The watershed line cannot be built by computing the The watershed line cannot be built by computing the trajectory of rain drops streaming on the topographic surface trajectory of rain drops streaming on the topographic surface (run off) (run off) FORGET ITFORGET IT
BIASES AND FALSITIES WITH THE WATERSHED (2)BIASES AND FALSITIES WITH THE WATERSHED (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 10
A watershed line is not a local feature In particular it is noA watershed line is not a local feature In particular it is not t linked to local structures (crest lines ridgeslinked to local structures (crest lines ridgeshelliphellip) The watershed is ) The watershed is not a LOCAL conceptnot a LOCAL conceptOne canOne canrsquorsquot with the only local knowledge of the neighborhood of a t with the only local knowledge of the neighborhood of a point answer the questionpoint answer the question
Does this point belong to the watershed lineDoes this point belong to the watershed line
BIASES AND FALSITIES WITH THE WTS (3)BIASES AND FALSITIES WITH THE WTS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 11
Is flooding always an ascending processIs flooding always an ascending processIn other words when flooding has reached altitude h is it trueIn other words when flooding has reached altitude h is it true that that ALL the points at a lower altitude have been floodedALL the points at a lower altitude have been flooded
The answer is NO CounterThe answer is NO Counter--example the buttonexample the button--holehole
BIASES AND FALSITIES WITH THE WATERSHED (4)BIASES AND FALSITIES WITH THE WATERSHED (4)
((truetrue buttonbutton--holehole structure in french structure in french GresivaudanGresivaudan))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 12
The watershed transform is used for image segmentationThe watershed transform is used for image segmentation
bullbull GreytoneGreytone segmentationsegmentation
bullbull Shape segmentationShape segmentation
Cutting objects into a union of ldquoconvexrdquo sets by means of the watershed of the distance function
The watershed of the gradient corresponds to the contour lines
USE OF WATERSHEDUSE OF WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 13
Morphological gradientMorphological gradient
g(f) = (f g(f) = (f B) B) -- (f (f 0 0 B)B)Other morphological gradients (half-gradients) can also be definedg-(f) = f - (f 0 B)g+(f) = (f B) ndash fThick gradientsggii(f(f) = (f ) = (f BBii) ) -- (f (f 00 BBii))Regularized gradientsRegularized gradients
THE GRADIENT A REMINDERTHE GRADIENT A REMINDER
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 14
The gradient watershed is overThe gradient watershed is over--segmentedsegmented
To avoid this overTo avoid this over--segmentation due segmentation due to numerous sources of flooding one to numerous sources of flooding one can select some of them (the markers) can select some of them (the markers) and perform the watershed transform and perform the watershed transform controlled by these markerscontrolled by these markers
Gradient images are often noisy and contain a large number of minima Each minimum generates a catchment basin in the WTS
THE MARKERTHE MARKER--CONTROLLED WATERSHEDCONTROLLED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 15
Road segmentationRoad segmentation
Original imageOriginal image
gradientgradient
markersmarkersMarkerMarker--controlled watershed of controlled watershed of
the gradientthe gradient
EXAMPLE OF MARKEREXAMPLE OF MARKER--CONTROLLED CONTROLLED WATERSHEDWATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 16
bullbull Level by level floodingLevel by level flooding
bullbull Hierarchical queuesHierarchical queues
W = M marker setW = M marker set00
W = SKIZ (W )W = SKIZ (W )ii ii--11Z (f) Z (f) 4 4 MMii
This algorithm is simpler than the classical one there is no minima detection
A token at level iltj (the current level) may appear In this case it is treated as a token at level j (the i-queue is no longer alive) With marker-controlled
watershed overflow is the rule and not the exception
MARKERMARKER--CONTROLLED WATERSHED CONTROLLED WATERSHED ALGORITHMSALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 5
bull ItItrsquorsquos a flooding processs a flooding processbull Flooding sources are theFlooding sources are the
minima of the functionminima of the function
Two hierarchies are usedTwo hierarchies are usedbullbull flood progression with theflood progression with the
altitude (sequential process)altitude (sequential process)bullbull flood on the plateausflat zonesflood on the plateausflat zones(parallel process)(parallel process)
The result is a partition of the The result is a partition of the image into catchment basins image into catchment basins and watershed lines (dams)and watershed lines (dams)
THE CLASSICAL WATERSHED ALGORITHMTHE CLASSICAL WATERSHED ALGORITHM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 6
The transformation can be The transformation can be expressed on the successive expressed on the successive levels Z of the function flevels Z of the function f
Use of the geodesic SKIZ Use of the geodesic SKIZ transform to simulate the transform to simulate the propagation without merging propagation without merging
ii
W = m (f)W = m (f)0000The catchment basins at level 0 are the minima at this level
W = W = [[ SKIZ (W )SKIZ (W )] ] 44 m (f) m (f) i+1i+1i+1i+1 Z (f)Z (f)i+1i+1
ii
with
m (f) = Z (f) R (Z (f))m (f) = Z (f) R (Z (f))
R is the geodesic reconstruction
i+1i+1 i+1i+1 Z (f)Z (f) iii+1i+1
THE CLASSICAL WATERSHED ALGORITHM (2)THE CLASSICAL WATERSHED ALGORITHM (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 7
bullbull Classical one (SKIZ using rotation thickenings)Classical one (SKIZ using rotation thickenings)
bullbull Hierarchical queues (a priori order defined in the queue)Hierarchical queues (a priori order defined in the queue)
The use of rotating structuring elements in the SKIZ generates a non isotropic flooding on the plateaus
The tokens in the same stack should be processed at the same time
Most of the watershed algorithms are biased
bullbull WatershedsWatersheds basedbased on graphson graphs
WATERSHED ALGORITHMSWATERSHED ALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 8
For various reasons (complexity computation speed lazinessFor various reasons (complexity computation speed lazinesshelliphellip) ) the unbiased watershed transforms are seldom usedthe unbiased watershed transforms are seldom used
Comparison between a true watershed (left) and the result of a ldquoclassicalrdquo algorithm
These biases may have dramatic consequences in hierarchical These biases may have dramatic consequences in hierarchical approaches based on the comparison of adjacent catchment basinsapproaches based on the comparison of adjacent catchment basins
BIASES AND FALSITIES WITH THE WATERSHEDBIASES AND FALSITIES WITH THE WATERSHED
Due to Due to thisthis biasbias the the watershedwatershed transformtransform isis not UNIQUE (not UNIQUE (shouldshouldbebe uniqueuniquehelliphellip) It ) It dependsdepends on the on the algorithmalgorithm usedused to to buildbuild itit
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 9
In any case the results could not be identical (due to the propagation on the flat zones)
The flooding on the plateaus is based on a MODEL (constant speed) It has mainly two advantages it is simple and it has a physical meaning
The watershed line cannot be built by computing the The watershed line cannot be built by computing the trajectory of rain drops streaming on the topographic surface trajectory of rain drops streaming on the topographic surface (run off) (run off) FORGET ITFORGET IT
BIASES AND FALSITIES WITH THE WATERSHED (2)BIASES AND FALSITIES WITH THE WATERSHED (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 10
A watershed line is not a local feature In particular it is noA watershed line is not a local feature In particular it is not t linked to local structures (crest lines ridgeslinked to local structures (crest lines ridgeshelliphellip) The watershed is ) The watershed is not a LOCAL conceptnot a LOCAL conceptOne canOne canrsquorsquot with the only local knowledge of the neighborhood of a t with the only local knowledge of the neighborhood of a point answer the questionpoint answer the question
Does this point belong to the watershed lineDoes this point belong to the watershed line
BIASES AND FALSITIES WITH THE WTS (3)BIASES AND FALSITIES WITH THE WTS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 11
Is flooding always an ascending processIs flooding always an ascending processIn other words when flooding has reached altitude h is it trueIn other words when flooding has reached altitude h is it true that that ALL the points at a lower altitude have been floodedALL the points at a lower altitude have been flooded
The answer is NO CounterThe answer is NO Counter--example the buttonexample the button--holehole
BIASES AND FALSITIES WITH THE WATERSHED (4)BIASES AND FALSITIES WITH THE WATERSHED (4)
((truetrue buttonbutton--holehole structure in french structure in french GresivaudanGresivaudan))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 12
The watershed transform is used for image segmentationThe watershed transform is used for image segmentation
bullbull GreytoneGreytone segmentationsegmentation
bullbull Shape segmentationShape segmentation
Cutting objects into a union of ldquoconvexrdquo sets by means of the watershed of the distance function
The watershed of the gradient corresponds to the contour lines
USE OF WATERSHEDUSE OF WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 13
Morphological gradientMorphological gradient
g(f) = (f g(f) = (f B) B) -- (f (f 0 0 B)B)Other morphological gradients (half-gradients) can also be definedg-(f) = f - (f 0 B)g+(f) = (f B) ndash fThick gradientsggii(f(f) = (f ) = (f BBii) ) -- (f (f 00 BBii))Regularized gradientsRegularized gradients
THE GRADIENT A REMINDERTHE GRADIENT A REMINDER
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 14
The gradient watershed is overThe gradient watershed is over--segmentedsegmented
To avoid this overTo avoid this over--segmentation due segmentation due to numerous sources of flooding one to numerous sources of flooding one can select some of them (the markers) can select some of them (the markers) and perform the watershed transform and perform the watershed transform controlled by these markerscontrolled by these markers
Gradient images are often noisy and contain a large number of minima Each minimum generates a catchment basin in the WTS
THE MARKERTHE MARKER--CONTROLLED WATERSHEDCONTROLLED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 15
Road segmentationRoad segmentation
Original imageOriginal image
gradientgradient
markersmarkersMarkerMarker--controlled watershed of controlled watershed of
the gradientthe gradient
EXAMPLE OF MARKEREXAMPLE OF MARKER--CONTROLLED CONTROLLED WATERSHEDWATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 16
bullbull Level by level floodingLevel by level flooding
bullbull Hierarchical queuesHierarchical queues
W = M marker setW = M marker set00
W = SKIZ (W )W = SKIZ (W )ii ii--11Z (f) Z (f) 4 4 MMii
This algorithm is simpler than the classical one there is no minima detection
A token at level iltj (the current level) may appear In this case it is treated as a token at level j (the i-queue is no longer alive) With marker-controlled
watershed overflow is the rule and not the exception
MARKERMARKER--CONTROLLED WATERSHED CONTROLLED WATERSHED ALGORITHMSALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 6
The transformation can be The transformation can be expressed on the successive expressed on the successive levels Z of the function flevels Z of the function f
Use of the geodesic SKIZ Use of the geodesic SKIZ transform to simulate the transform to simulate the propagation without merging propagation without merging
ii
W = m (f)W = m (f)0000The catchment basins at level 0 are the minima at this level
W = W = [[ SKIZ (W )SKIZ (W )] ] 44 m (f) m (f) i+1i+1i+1i+1 Z (f)Z (f)i+1i+1
ii
with
m (f) = Z (f) R (Z (f))m (f) = Z (f) R (Z (f))
R is the geodesic reconstruction
i+1i+1 i+1i+1 Z (f)Z (f) iii+1i+1
THE CLASSICAL WATERSHED ALGORITHM (2)THE CLASSICAL WATERSHED ALGORITHM (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 7
bullbull Classical one (SKIZ using rotation thickenings)Classical one (SKIZ using rotation thickenings)
bullbull Hierarchical queues (a priori order defined in the queue)Hierarchical queues (a priori order defined in the queue)
The use of rotating structuring elements in the SKIZ generates a non isotropic flooding on the plateaus
The tokens in the same stack should be processed at the same time
Most of the watershed algorithms are biased
bullbull WatershedsWatersheds basedbased on graphson graphs
WATERSHED ALGORITHMSWATERSHED ALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 8
For various reasons (complexity computation speed lazinessFor various reasons (complexity computation speed lazinesshelliphellip) ) the unbiased watershed transforms are seldom usedthe unbiased watershed transforms are seldom used
Comparison between a true watershed (left) and the result of a ldquoclassicalrdquo algorithm
These biases may have dramatic consequences in hierarchical These biases may have dramatic consequences in hierarchical approaches based on the comparison of adjacent catchment basinsapproaches based on the comparison of adjacent catchment basins
BIASES AND FALSITIES WITH THE WATERSHEDBIASES AND FALSITIES WITH THE WATERSHED
Due to Due to thisthis biasbias the the watershedwatershed transformtransform isis not UNIQUE (not UNIQUE (shouldshouldbebe uniqueuniquehelliphellip) It ) It dependsdepends on the on the algorithmalgorithm usedused to to buildbuild itit
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 9
In any case the results could not be identical (due to the propagation on the flat zones)
The flooding on the plateaus is based on a MODEL (constant speed) It has mainly two advantages it is simple and it has a physical meaning
The watershed line cannot be built by computing the The watershed line cannot be built by computing the trajectory of rain drops streaming on the topographic surface trajectory of rain drops streaming on the topographic surface (run off) (run off) FORGET ITFORGET IT
BIASES AND FALSITIES WITH THE WATERSHED (2)BIASES AND FALSITIES WITH THE WATERSHED (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 10
A watershed line is not a local feature In particular it is noA watershed line is not a local feature In particular it is not t linked to local structures (crest lines ridgeslinked to local structures (crest lines ridgeshelliphellip) The watershed is ) The watershed is not a LOCAL conceptnot a LOCAL conceptOne canOne canrsquorsquot with the only local knowledge of the neighborhood of a t with the only local knowledge of the neighborhood of a point answer the questionpoint answer the question
Does this point belong to the watershed lineDoes this point belong to the watershed line
BIASES AND FALSITIES WITH THE WTS (3)BIASES AND FALSITIES WITH THE WTS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 11
Is flooding always an ascending processIs flooding always an ascending processIn other words when flooding has reached altitude h is it trueIn other words when flooding has reached altitude h is it true that that ALL the points at a lower altitude have been floodedALL the points at a lower altitude have been flooded
The answer is NO CounterThe answer is NO Counter--example the buttonexample the button--holehole
BIASES AND FALSITIES WITH THE WATERSHED (4)BIASES AND FALSITIES WITH THE WATERSHED (4)
((truetrue buttonbutton--holehole structure in french structure in french GresivaudanGresivaudan))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 12
The watershed transform is used for image segmentationThe watershed transform is used for image segmentation
bullbull GreytoneGreytone segmentationsegmentation
bullbull Shape segmentationShape segmentation
Cutting objects into a union of ldquoconvexrdquo sets by means of the watershed of the distance function
The watershed of the gradient corresponds to the contour lines
USE OF WATERSHEDUSE OF WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 13
Morphological gradientMorphological gradient
g(f) = (f g(f) = (f B) B) -- (f (f 0 0 B)B)Other morphological gradients (half-gradients) can also be definedg-(f) = f - (f 0 B)g+(f) = (f B) ndash fThick gradientsggii(f(f) = (f ) = (f BBii) ) -- (f (f 00 BBii))Regularized gradientsRegularized gradients
THE GRADIENT A REMINDERTHE GRADIENT A REMINDER
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 14
The gradient watershed is overThe gradient watershed is over--segmentedsegmented
To avoid this overTo avoid this over--segmentation due segmentation due to numerous sources of flooding one to numerous sources of flooding one can select some of them (the markers) can select some of them (the markers) and perform the watershed transform and perform the watershed transform controlled by these markerscontrolled by these markers
Gradient images are often noisy and contain a large number of minima Each minimum generates a catchment basin in the WTS
THE MARKERTHE MARKER--CONTROLLED WATERSHEDCONTROLLED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 15
Road segmentationRoad segmentation
Original imageOriginal image
gradientgradient
markersmarkersMarkerMarker--controlled watershed of controlled watershed of
the gradientthe gradient
EXAMPLE OF MARKEREXAMPLE OF MARKER--CONTROLLED CONTROLLED WATERSHEDWATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 16
bullbull Level by level floodingLevel by level flooding
bullbull Hierarchical queuesHierarchical queues
W = M marker setW = M marker set00
W = SKIZ (W )W = SKIZ (W )ii ii--11Z (f) Z (f) 4 4 MMii
This algorithm is simpler than the classical one there is no minima detection
A token at level iltj (the current level) may appear In this case it is treated as a token at level j (the i-queue is no longer alive) With marker-controlled
watershed overflow is the rule and not the exception
MARKERMARKER--CONTROLLED WATERSHED CONTROLLED WATERSHED ALGORITHMSALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 7
bullbull Classical one (SKIZ using rotation thickenings)Classical one (SKIZ using rotation thickenings)
bullbull Hierarchical queues (a priori order defined in the queue)Hierarchical queues (a priori order defined in the queue)
The use of rotating structuring elements in the SKIZ generates a non isotropic flooding on the plateaus
The tokens in the same stack should be processed at the same time
Most of the watershed algorithms are biased
bullbull WatershedsWatersheds basedbased on graphson graphs
WATERSHED ALGORITHMSWATERSHED ALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 8
For various reasons (complexity computation speed lazinessFor various reasons (complexity computation speed lazinesshelliphellip) ) the unbiased watershed transforms are seldom usedthe unbiased watershed transforms are seldom used
Comparison between a true watershed (left) and the result of a ldquoclassicalrdquo algorithm
These biases may have dramatic consequences in hierarchical These biases may have dramatic consequences in hierarchical approaches based on the comparison of adjacent catchment basinsapproaches based on the comparison of adjacent catchment basins
BIASES AND FALSITIES WITH THE WATERSHEDBIASES AND FALSITIES WITH THE WATERSHED
Due to Due to thisthis biasbias the the watershedwatershed transformtransform isis not UNIQUE (not UNIQUE (shouldshouldbebe uniqueuniquehelliphellip) It ) It dependsdepends on the on the algorithmalgorithm usedused to to buildbuild itit
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 9
In any case the results could not be identical (due to the propagation on the flat zones)
The flooding on the plateaus is based on a MODEL (constant speed) It has mainly two advantages it is simple and it has a physical meaning
The watershed line cannot be built by computing the The watershed line cannot be built by computing the trajectory of rain drops streaming on the topographic surface trajectory of rain drops streaming on the topographic surface (run off) (run off) FORGET ITFORGET IT
BIASES AND FALSITIES WITH THE WATERSHED (2)BIASES AND FALSITIES WITH THE WATERSHED (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 10
A watershed line is not a local feature In particular it is noA watershed line is not a local feature In particular it is not t linked to local structures (crest lines ridgeslinked to local structures (crest lines ridgeshelliphellip) The watershed is ) The watershed is not a LOCAL conceptnot a LOCAL conceptOne canOne canrsquorsquot with the only local knowledge of the neighborhood of a t with the only local knowledge of the neighborhood of a point answer the questionpoint answer the question
Does this point belong to the watershed lineDoes this point belong to the watershed line
BIASES AND FALSITIES WITH THE WTS (3)BIASES AND FALSITIES WITH THE WTS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 11
Is flooding always an ascending processIs flooding always an ascending processIn other words when flooding has reached altitude h is it trueIn other words when flooding has reached altitude h is it true that that ALL the points at a lower altitude have been floodedALL the points at a lower altitude have been flooded
The answer is NO CounterThe answer is NO Counter--example the buttonexample the button--holehole
BIASES AND FALSITIES WITH THE WATERSHED (4)BIASES AND FALSITIES WITH THE WATERSHED (4)
((truetrue buttonbutton--holehole structure in french structure in french GresivaudanGresivaudan))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 12
The watershed transform is used for image segmentationThe watershed transform is used for image segmentation
bullbull GreytoneGreytone segmentationsegmentation
bullbull Shape segmentationShape segmentation
Cutting objects into a union of ldquoconvexrdquo sets by means of the watershed of the distance function
The watershed of the gradient corresponds to the contour lines
USE OF WATERSHEDUSE OF WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 13
Morphological gradientMorphological gradient
g(f) = (f g(f) = (f B) B) -- (f (f 0 0 B)B)Other morphological gradients (half-gradients) can also be definedg-(f) = f - (f 0 B)g+(f) = (f B) ndash fThick gradientsggii(f(f) = (f ) = (f BBii) ) -- (f (f 00 BBii))Regularized gradientsRegularized gradients
THE GRADIENT A REMINDERTHE GRADIENT A REMINDER
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 14
The gradient watershed is overThe gradient watershed is over--segmentedsegmented
To avoid this overTo avoid this over--segmentation due segmentation due to numerous sources of flooding one to numerous sources of flooding one can select some of them (the markers) can select some of them (the markers) and perform the watershed transform and perform the watershed transform controlled by these markerscontrolled by these markers
Gradient images are often noisy and contain a large number of minima Each minimum generates a catchment basin in the WTS
THE MARKERTHE MARKER--CONTROLLED WATERSHEDCONTROLLED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 15
Road segmentationRoad segmentation
Original imageOriginal image
gradientgradient
markersmarkersMarkerMarker--controlled watershed of controlled watershed of
the gradientthe gradient
EXAMPLE OF MARKEREXAMPLE OF MARKER--CONTROLLED CONTROLLED WATERSHEDWATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 16
bullbull Level by level floodingLevel by level flooding
bullbull Hierarchical queuesHierarchical queues
W = M marker setW = M marker set00
W = SKIZ (W )W = SKIZ (W )ii ii--11Z (f) Z (f) 4 4 MMii
This algorithm is simpler than the classical one there is no minima detection
A token at level iltj (the current level) may appear In this case it is treated as a token at level j (the i-queue is no longer alive) With marker-controlled
watershed overflow is the rule and not the exception
MARKERMARKER--CONTROLLED WATERSHED CONTROLLED WATERSHED ALGORITHMSALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 8
For various reasons (complexity computation speed lazinessFor various reasons (complexity computation speed lazinesshelliphellip) ) the unbiased watershed transforms are seldom usedthe unbiased watershed transforms are seldom used
Comparison between a true watershed (left) and the result of a ldquoclassicalrdquo algorithm
These biases may have dramatic consequences in hierarchical These biases may have dramatic consequences in hierarchical approaches based on the comparison of adjacent catchment basinsapproaches based on the comparison of adjacent catchment basins
BIASES AND FALSITIES WITH THE WATERSHEDBIASES AND FALSITIES WITH THE WATERSHED
Due to Due to thisthis biasbias the the watershedwatershed transformtransform isis not UNIQUE (not UNIQUE (shouldshouldbebe uniqueuniquehelliphellip) It ) It dependsdepends on the on the algorithmalgorithm usedused to to buildbuild itit
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 9
In any case the results could not be identical (due to the propagation on the flat zones)
The flooding on the plateaus is based on a MODEL (constant speed) It has mainly two advantages it is simple and it has a physical meaning
The watershed line cannot be built by computing the The watershed line cannot be built by computing the trajectory of rain drops streaming on the topographic surface trajectory of rain drops streaming on the topographic surface (run off) (run off) FORGET ITFORGET IT
BIASES AND FALSITIES WITH THE WATERSHED (2)BIASES AND FALSITIES WITH THE WATERSHED (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 10
A watershed line is not a local feature In particular it is noA watershed line is not a local feature In particular it is not t linked to local structures (crest lines ridgeslinked to local structures (crest lines ridgeshelliphellip) The watershed is ) The watershed is not a LOCAL conceptnot a LOCAL conceptOne canOne canrsquorsquot with the only local knowledge of the neighborhood of a t with the only local knowledge of the neighborhood of a point answer the questionpoint answer the question
Does this point belong to the watershed lineDoes this point belong to the watershed line
BIASES AND FALSITIES WITH THE WTS (3)BIASES AND FALSITIES WITH THE WTS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 11
Is flooding always an ascending processIs flooding always an ascending processIn other words when flooding has reached altitude h is it trueIn other words when flooding has reached altitude h is it true that that ALL the points at a lower altitude have been floodedALL the points at a lower altitude have been flooded
The answer is NO CounterThe answer is NO Counter--example the buttonexample the button--holehole
BIASES AND FALSITIES WITH THE WATERSHED (4)BIASES AND FALSITIES WITH THE WATERSHED (4)
((truetrue buttonbutton--holehole structure in french structure in french GresivaudanGresivaudan))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 12
The watershed transform is used for image segmentationThe watershed transform is used for image segmentation
bullbull GreytoneGreytone segmentationsegmentation
bullbull Shape segmentationShape segmentation
Cutting objects into a union of ldquoconvexrdquo sets by means of the watershed of the distance function
The watershed of the gradient corresponds to the contour lines
USE OF WATERSHEDUSE OF WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 13
Morphological gradientMorphological gradient
g(f) = (f g(f) = (f B) B) -- (f (f 0 0 B)B)Other morphological gradients (half-gradients) can also be definedg-(f) = f - (f 0 B)g+(f) = (f B) ndash fThick gradientsggii(f(f) = (f ) = (f BBii) ) -- (f (f 00 BBii))Regularized gradientsRegularized gradients
THE GRADIENT A REMINDERTHE GRADIENT A REMINDER
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 14
The gradient watershed is overThe gradient watershed is over--segmentedsegmented
To avoid this overTo avoid this over--segmentation due segmentation due to numerous sources of flooding one to numerous sources of flooding one can select some of them (the markers) can select some of them (the markers) and perform the watershed transform and perform the watershed transform controlled by these markerscontrolled by these markers
Gradient images are often noisy and contain a large number of minima Each minimum generates a catchment basin in the WTS
THE MARKERTHE MARKER--CONTROLLED WATERSHEDCONTROLLED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 15
Road segmentationRoad segmentation
Original imageOriginal image
gradientgradient
markersmarkersMarkerMarker--controlled watershed of controlled watershed of
the gradientthe gradient
EXAMPLE OF MARKEREXAMPLE OF MARKER--CONTROLLED CONTROLLED WATERSHEDWATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 16
bullbull Level by level floodingLevel by level flooding
bullbull Hierarchical queuesHierarchical queues
W = M marker setW = M marker set00
W = SKIZ (W )W = SKIZ (W )ii ii--11Z (f) Z (f) 4 4 MMii
This algorithm is simpler than the classical one there is no minima detection
A token at level iltj (the current level) may appear In this case it is treated as a token at level j (the i-queue is no longer alive) With marker-controlled
watershed overflow is the rule and not the exception
MARKERMARKER--CONTROLLED WATERSHED CONTROLLED WATERSHED ALGORITHMSALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 9
In any case the results could not be identical (due to the propagation on the flat zones)
The flooding on the plateaus is based on a MODEL (constant speed) It has mainly two advantages it is simple and it has a physical meaning
The watershed line cannot be built by computing the The watershed line cannot be built by computing the trajectory of rain drops streaming on the topographic surface trajectory of rain drops streaming on the topographic surface (run off) (run off) FORGET ITFORGET IT
BIASES AND FALSITIES WITH THE WATERSHED (2)BIASES AND FALSITIES WITH THE WATERSHED (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 10
A watershed line is not a local feature In particular it is noA watershed line is not a local feature In particular it is not t linked to local structures (crest lines ridgeslinked to local structures (crest lines ridgeshelliphellip) The watershed is ) The watershed is not a LOCAL conceptnot a LOCAL conceptOne canOne canrsquorsquot with the only local knowledge of the neighborhood of a t with the only local knowledge of the neighborhood of a point answer the questionpoint answer the question
Does this point belong to the watershed lineDoes this point belong to the watershed line
BIASES AND FALSITIES WITH THE WTS (3)BIASES AND FALSITIES WITH THE WTS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 11
Is flooding always an ascending processIs flooding always an ascending processIn other words when flooding has reached altitude h is it trueIn other words when flooding has reached altitude h is it true that that ALL the points at a lower altitude have been floodedALL the points at a lower altitude have been flooded
The answer is NO CounterThe answer is NO Counter--example the buttonexample the button--holehole
BIASES AND FALSITIES WITH THE WATERSHED (4)BIASES AND FALSITIES WITH THE WATERSHED (4)
((truetrue buttonbutton--holehole structure in french structure in french GresivaudanGresivaudan))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 12
The watershed transform is used for image segmentationThe watershed transform is used for image segmentation
bullbull GreytoneGreytone segmentationsegmentation
bullbull Shape segmentationShape segmentation
Cutting objects into a union of ldquoconvexrdquo sets by means of the watershed of the distance function
The watershed of the gradient corresponds to the contour lines
USE OF WATERSHEDUSE OF WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 13
Morphological gradientMorphological gradient
g(f) = (f g(f) = (f B) B) -- (f (f 0 0 B)B)Other morphological gradients (half-gradients) can also be definedg-(f) = f - (f 0 B)g+(f) = (f B) ndash fThick gradientsggii(f(f) = (f ) = (f BBii) ) -- (f (f 00 BBii))Regularized gradientsRegularized gradients
THE GRADIENT A REMINDERTHE GRADIENT A REMINDER
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 14
The gradient watershed is overThe gradient watershed is over--segmentedsegmented
To avoid this overTo avoid this over--segmentation due segmentation due to numerous sources of flooding one to numerous sources of flooding one can select some of them (the markers) can select some of them (the markers) and perform the watershed transform and perform the watershed transform controlled by these markerscontrolled by these markers
Gradient images are often noisy and contain a large number of minima Each minimum generates a catchment basin in the WTS
THE MARKERTHE MARKER--CONTROLLED WATERSHEDCONTROLLED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 15
Road segmentationRoad segmentation
Original imageOriginal image
gradientgradient
markersmarkersMarkerMarker--controlled watershed of controlled watershed of
the gradientthe gradient
EXAMPLE OF MARKEREXAMPLE OF MARKER--CONTROLLED CONTROLLED WATERSHEDWATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 16
bullbull Level by level floodingLevel by level flooding
bullbull Hierarchical queuesHierarchical queues
W = M marker setW = M marker set00
W = SKIZ (W )W = SKIZ (W )ii ii--11Z (f) Z (f) 4 4 MMii
This algorithm is simpler than the classical one there is no minima detection
A token at level iltj (the current level) may appear In this case it is treated as a token at level j (the i-queue is no longer alive) With marker-controlled
watershed overflow is the rule and not the exception
MARKERMARKER--CONTROLLED WATERSHED CONTROLLED WATERSHED ALGORITHMSALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 10
A watershed line is not a local feature In particular it is noA watershed line is not a local feature In particular it is not t linked to local structures (crest lines ridgeslinked to local structures (crest lines ridgeshelliphellip) The watershed is ) The watershed is not a LOCAL conceptnot a LOCAL conceptOne canOne canrsquorsquot with the only local knowledge of the neighborhood of a t with the only local knowledge of the neighborhood of a point answer the questionpoint answer the question
Does this point belong to the watershed lineDoes this point belong to the watershed line
BIASES AND FALSITIES WITH THE WTS (3)BIASES AND FALSITIES WITH THE WTS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 11
Is flooding always an ascending processIs flooding always an ascending processIn other words when flooding has reached altitude h is it trueIn other words when flooding has reached altitude h is it true that that ALL the points at a lower altitude have been floodedALL the points at a lower altitude have been flooded
The answer is NO CounterThe answer is NO Counter--example the buttonexample the button--holehole
BIASES AND FALSITIES WITH THE WATERSHED (4)BIASES AND FALSITIES WITH THE WATERSHED (4)
((truetrue buttonbutton--holehole structure in french structure in french GresivaudanGresivaudan))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 12
The watershed transform is used for image segmentationThe watershed transform is used for image segmentation
bullbull GreytoneGreytone segmentationsegmentation
bullbull Shape segmentationShape segmentation
Cutting objects into a union of ldquoconvexrdquo sets by means of the watershed of the distance function
The watershed of the gradient corresponds to the contour lines
USE OF WATERSHEDUSE OF WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 13
Morphological gradientMorphological gradient
g(f) = (f g(f) = (f B) B) -- (f (f 0 0 B)B)Other morphological gradients (half-gradients) can also be definedg-(f) = f - (f 0 B)g+(f) = (f B) ndash fThick gradientsggii(f(f) = (f ) = (f BBii) ) -- (f (f 00 BBii))Regularized gradientsRegularized gradients
THE GRADIENT A REMINDERTHE GRADIENT A REMINDER
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 14
The gradient watershed is overThe gradient watershed is over--segmentedsegmented
To avoid this overTo avoid this over--segmentation due segmentation due to numerous sources of flooding one to numerous sources of flooding one can select some of them (the markers) can select some of them (the markers) and perform the watershed transform and perform the watershed transform controlled by these markerscontrolled by these markers
Gradient images are often noisy and contain a large number of minima Each minimum generates a catchment basin in the WTS
THE MARKERTHE MARKER--CONTROLLED WATERSHEDCONTROLLED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 15
Road segmentationRoad segmentation
Original imageOriginal image
gradientgradient
markersmarkersMarkerMarker--controlled watershed of controlled watershed of
the gradientthe gradient
EXAMPLE OF MARKEREXAMPLE OF MARKER--CONTROLLED CONTROLLED WATERSHEDWATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 16
bullbull Level by level floodingLevel by level flooding
bullbull Hierarchical queuesHierarchical queues
W = M marker setW = M marker set00
W = SKIZ (W )W = SKIZ (W )ii ii--11Z (f) Z (f) 4 4 MMii
This algorithm is simpler than the classical one there is no minima detection
A token at level iltj (the current level) may appear In this case it is treated as a token at level j (the i-queue is no longer alive) With marker-controlled
watershed overflow is the rule and not the exception
MARKERMARKER--CONTROLLED WATERSHED CONTROLLED WATERSHED ALGORITHMSALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 11
Is flooding always an ascending processIs flooding always an ascending processIn other words when flooding has reached altitude h is it trueIn other words when flooding has reached altitude h is it true that that ALL the points at a lower altitude have been floodedALL the points at a lower altitude have been flooded
The answer is NO CounterThe answer is NO Counter--example the buttonexample the button--holehole
BIASES AND FALSITIES WITH THE WATERSHED (4)BIASES AND FALSITIES WITH THE WATERSHED (4)
((truetrue buttonbutton--holehole structure in french structure in french GresivaudanGresivaudan))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 12
The watershed transform is used for image segmentationThe watershed transform is used for image segmentation
bullbull GreytoneGreytone segmentationsegmentation
bullbull Shape segmentationShape segmentation
Cutting objects into a union of ldquoconvexrdquo sets by means of the watershed of the distance function
The watershed of the gradient corresponds to the contour lines
USE OF WATERSHEDUSE OF WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 13
Morphological gradientMorphological gradient
g(f) = (f g(f) = (f B) B) -- (f (f 0 0 B)B)Other morphological gradients (half-gradients) can also be definedg-(f) = f - (f 0 B)g+(f) = (f B) ndash fThick gradientsggii(f(f) = (f ) = (f BBii) ) -- (f (f 00 BBii))Regularized gradientsRegularized gradients
THE GRADIENT A REMINDERTHE GRADIENT A REMINDER
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 14
The gradient watershed is overThe gradient watershed is over--segmentedsegmented
To avoid this overTo avoid this over--segmentation due segmentation due to numerous sources of flooding one to numerous sources of flooding one can select some of them (the markers) can select some of them (the markers) and perform the watershed transform and perform the watershed transform controlled by these markerscontrolled by these markers
Gradient images are often noisy and contain a large number of minima Each minimum generates a catchment basin in the WTS
THE MARKERTHE MARKER--CONTROLLED WATERSHEDCONTROLLED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 15
Road segmentationRoad segmentation
Original imageOriginal image
gradientgradient
markersmarkersMarkerMarker--controlled watershed of controlled watershed of
the gradientthe gradient
EXAMPLE OF MARKEREXAMPLE OF MARKER--CONTROLLED CONTROLLED WATERSHEDWATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 16
bullbull Level by level floodingLevel by level flooding
bullbull Hierarchical queuesHierarchical queues
W = M marker setW = M marker set00
W = SKIZ (W )W = SKIZ (W )ii ii--11Z (f) Z (f) 4 4 MMii
This algorithm is simpler than the classical one there is no minima detection
A token at level iltj (the current level) may appear In this case it is treated as a token at level j (the i-queue is no longer alive) With marker-controlled
watershed overflow is the rule and not the exception
MARKERMARKER--CONTROLLED WATERSHED CONTROLLED WATERSHED ALGORITHMSALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 12
The watershed transform is used for image segmentationThe watershed transform is used for image segmentation
bullbull GreytoneGreytone segmentationsegmentation
bullbull Shape segmentationShape segmentation
Cutting objects into a union of ldquoconvexrdquo sets by means of the watershed of the distance function
The watershed of the gradient corresponds to the contour lines
USE OF WATERSHEDUSE OF WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 13
Morphological gradientMorphological gradient
g(f) = (f g(f) = (f B) B) -- (f (f 0 0 B)B)Other morphological gradients (half-gradients) can also be definedg-(f) = f - (f 0 B)g+(f) = (f B) ndash fThick gradientsggii(f(f) = (f ) = (f BBii) ) -- (f (f 00 BBii))Regularized gradientsRegularized gradients
THE GRADIENT A REMINDERTHE GRADIENT A REMINDER
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 14
The gradient watershed is overThe gradient watershed is over--segmentedsegmented
To avoid this overTo avoid this over--segmentation due segmentation due to numerous sources of flooding one to numerous sources of flooding one can select some of them (the markers) can select some of them (the markers) and perform the watershed transform and perform the watershed transform controlled by these markerscontrolled by these markers
Gradient images are often noisy and contain a large number of minima Each minimum generates a catchment basin in the WTS
THE MARKERTHE MARKER--CONTROLLED WATERSHEDCONTROLLED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 15
Road segmentationRoad segmentation
Original imageOriginal image
gradientgradient
markersmarkersMarkerMarker--controlled watershed of controlled watershed of
the gradientthe gradient
EXAMPLE OF MARKEREXAMPLE OF MARKER--CONTROLLED CONTROLLED WATERSHEDWATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 16
bullbull Level by level floodingLevel by level flooding
bullbull Hierarchical queuesHierarchical queues
W = M marker setW = M marker set00
W = SKIZ (W )W = SKIZ (W )ii ii--11Z (f) Z (f) 4 4 MMii
This algorithm is simpler than the classical one there is no minima detection
A token at level iltj (the current level) may appear In this case it is treated as a token at level j (the i-queue is no longer alive) With marker-controlled
watershed overflow is the rule and not the exception
MARKERMARKER--CONTROLLED WATERSHED CONTROLLED WATERSHED ALGORITHMSALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 13
Morphological gradientMorphological gradient
g(f) = (f g(f) = (f B) B) -- (f (f 0 0 B)B)Other morphological gradients (half-gradients) can also be definedg-(f) = f - (f 0 B)g+(f) = (f B) ndash fThick gradientsggii(f(f) = (f ) = (f BBii) ) -- (f (f 00 BBii))Regularized gradientsRegularized gradients
THE GRADIENT A REMINDERTHE GRADIENT A REMINDER
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 14
The gradient watershed is overThe gradient watershed is over--segmentedsegmented
To avoid this overTo avoid this over--segmentation due segmentation due to numerous sources of flooding one to numerous sources of flooding one can select some of them (the markers) can select some of them (the markers) and perform the watershed transform and perform the watershed transform controlled by these markerscontrolled by these markers
Gradient images are often noisy and contain a large number of minima Each minimum generates a catchment basin in the WTS
THE MARKERTHE MARKER--CONTROLLED WATERSHEDCONTROLLED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 15
Road segmentationRoad segmentation
Original imageOriginal image
gradientgradient
markersmarkersMarkerMarker--controlled watershed of controlled watershed of
the gradientthe gradient
EXAMPLE OF MARKEREXAMPLE OF MARKER--CONTROLLED CONTROLLED WATERSHEDWATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 16
bullbull Level by level floodingLevel by level flooding
bullbull Hierarchical queuesHierarchical queues
W = M marker setW = M marker set00
W = SKIZ (W )W = SKIZ (W )ii ii--11Z (f) Z (f) 4 4 MMii
This algorithm is simpler than the classical one there is no minima detection
A token at level iltj (the current level) may appear In this case it is treated as a token at level j (the i-queue is no longer alive) With marker-controlled
watershed overflow is the rule and not the exception
MARKERMARKER--CONTROLLED WATERSHED CONTROLLED WATERSHED ALGORITHMSALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 14
The gradient watershed is overThe gradient watershed is over--segmentedsegmented
To avoid this overTo avoid this over--segmentation due segmentation due to numerous sources of flooding one to numerous sources of flooding one can select some of them (the markers) can select some of them (the markers) and perform the watershed transform and perform the watershed transform controlled by these markerscontrolled by these markers
Gradient images are often noisy and contain a large number of minima Each minimum generates a catchment basin in the WTS
THE MARKERTHE MARKER--CONTROLLED WATERSHEDCONTROLLED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 15
Road segmentationRoad segmentation
Original imageOriginal image
gradientgradient
markersmarkersMarkerMarker--controlled watershed of controlled watershed of
the gradientthe gradient
EXAMPLE OF MARKEREXAMPLE OF MARKER--CONTROLLED CONTROLLED WATERSHEDWATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 16
bullbull Level by level floodingLevel by level flooding
bullbull Hierarchical queuesHierarchical queues
W = M marker setW = M marker set00
W = SKIZ (W )W = SKIZ (W )ii ii--11Z (f) Z (f) 4 4 MMii
This algorithm is simpler than the classical one there is no minima detection
A token at level iltj (the current level) may appear In this case it is treated as a token at level j (the i-queue is no longer alive) With marker-controlled
watershed overflow is the rule and not the exception
MARKERMARKER--CONTROLLED WATERSHED CONTROLLED WATERSHED ALGORITHMSALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 15
Road segmentationRoad segmentation
Original imageOriginal image
gradientgradient
markersmarkersMarkerMarker--controlled watershed of controlled watershed of
the gradientthe gradient
EXAMPLE OF MARKEREXAMPLE OF MARKER--CONTROLLED CONTROLLED WATERSHEDWATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 16
bullbull Level by level floodingLevel by level flooding
bullbull Hierarchical queuesHierarchical queues
W = M marker setW = M marker set00
W = SKIZ (W )W = SKIZ (W )ii ii--11Z (f) Z (f) 4 4 MMii
This algorithm is simpler than the classical one there is no minima detection
A token at level iltj (the current level) may appear In this case it is treated as a token at level j (the i-queue is no longer alive) With marker-controlled
watershed overflow is the rule and not the exception
MARKERMARKER--CONTROLLED WATERSHED CONTROLLED WATERSHED ALGORITHMSALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 16
bullbull Level by level floodingLevel by level flooding
bullbull Hierarchical queuesHierarchical queues
W = M marker setW = M marker set00
W = SKIZ (W )W = SKIZ (W )ii ii--11Z (f) Z (f) 4 4 MMii
This algorithm is simpler than the classical one there is no minima detection
A token at level iltj (the current level) may appear In this case it is treated as a token at level j (the i-queue is no longer alive) With marker-controlled
watershed overflow is the rule and not the exception
MARKERMARKER--CONTROLLED WATERSHED CONTROLLED WATERSHED ALGORITHMSALGORITHMS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 17
The geodesic reconstruction is widely used in mathematical The geodesic reconstruction is widely used in mathematical morphologymorphologybullbull detection of detection of extremaextrema (minima maxima)(minima maxima)bullbull filtering (openings and closings by reconstruction)filtering (openings and closings by reconstruction)bullbull watersheds (swamping watersheds (swamping homotopyhomotopy modification)modification)bullbull waterfallswaterfalls
gg ff R(f)R(f)gg
GEODESIC RECONSTRUCTIONGEODESIC RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 18
The geodesic reconstruction The geodesic reconstruction is of outmost importance for is of outmost importance for performing and performing and understanding the watershed understanding the watershed transformationtransformation
A dual reconstruction can A dual reconstruction can also be defined (it uses also be defined (it uses geodesic erosions)geodesic erosions)
GEODESIC RECONSTRUCTION (2)GEODESIC RECONSTRUCTION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 19
Based on reconstruction swamping allows to build a new functionBased on reconstruction swamping allows to build a new functionwhose minima correspond to the markerswhose minima correspond to the markers
1) a marker function is defined1) a marker function is definedh(xh(x) = ) = --1 1 iffiff x x isinisin MMh(xh(x) = g if not) = g if notmaxmax
2) The reconstruction of h over 2) The reconstruction of h over ggrsquorsquo = = inf(ghinf(gh) is made) is madeR (h) swamped functionR (h) swamped functionggrsquorsquo
SWAMPING (HOMOTOPY MODIFICATION)SWAMPING (HOMOTOPY MODIFICATION)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 20
When minima are replaced When minima are replaced by markers it is of outmost by markers it is of outmost importance to control the importance to control the position of these markersposition of these markers
Segmentation obtained (right image) with a marker-controlled watershed of the gradient (markers on the left)
POSITION OF THE MARKERSPOSITION OF THE MARKERS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 21
QuestionQuestionif we replace the original minima by if we replace the original minima by markers where to put the markers to markers where to put the markers to insure that the final watershed will be insure that the final watershed will be the samethe same
Notion of lower catchment basinNotion of lower catchment basinItrsquos the part of the catchment basin flooded before the first overflow (by the lower overflow zone)
Saddle zone
Solution the markers must be included in the lower catchment Solution the markers must be included in the lower catchment basins basins A one-to-one correspondence is not required provided that all the markers included in one lower catchment basin are given the same label
POSITION OF THE MARKERS (2)POSITION OF THE MARKERS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 22
THE SEGMENTATION PARADIGMTHE SEGMENTATION PARADIGM
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 23
WHICH CRITERIAWHICH CRITERIA
bullbull ContrastContrast criteriacriteriaGradientGradientTopTop--hathat transformtransform
bullbull Size and Size and shapeshape criteriacriteriaDistance Distance functionfunctionGranulometricGranulometric functionfunctionQuasiQuasi--distancedistance
bullbull CombinationCombination of of bothboth criteriacriteria ()()
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 24
Coffee grainsCoffee grains
The distance function of the set is computed This distance function is inverted and its watershed is performed The marker set is made of the maxima of the distance function
The watershed is performed on the support of the distance function The maxima are filtered to avoid over-segmentation due to parity problems
APPLICATIONSAPPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 25
Silver nitrate grains on a filmSilver nitrate grains on a film
Problem segmentation of the grains even when overlapping
Mask of the grains
1st markers maxima of distance function
Original image
Watershed of the distance function
2nd marker set
The background marker is added Final marker set
APPLICATIONS (2)APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 26
Original Filtered image
WatershedFinal result
Gradient
Markers
APPLICATIONS (3)APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 27
3D restitution of water drops from an hologram3D restitution of water drops from an hologram
bullbull n sections sn sections sbullbull find the best contourfind the best contourbullbull position x y z of each dropposition x y z of each dropbullbull volume volume
ii
A 3D image of an aerosol (artificial fog) is generated from an hologram Various sections of the 3D image are taken with a low focus depth camera
APPLICATIONS (4)APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 28
CriterionCriterionSup of the gradientsSup of the gradients
APPLICATIONS (5)APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 29
MarkersMarkers
bullbull Drops significant maxima of the filtered sup of all theDrops significant maxima of the filtered sup of all thesectionssections
bullbull Background watershed of Background watershed of the sup imagethe sup image(inverted)(inverted)
This watershed is a marker-controlled watershed (markers of the watershed are the drop markers)
APPLICATIONS (6)APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 30
Final watershed (left) The same watershed image superimposed on the different sections (right)
To find the best section a marker-controlled gradient watershed is performed on each section with the same set of markers (result in blue) and the best fit with the previous contour is determined The corresponding section gives the z-position of the drop
APPLICATIONS (7)APPLICATIONS (7)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 31
Traffic lanes segmentationTraffic lanes segmentation
From a sequence of n images f two images are computedbull The mean
f n
bull The mean of absolute differences|f -f |n
i
ji
i
The markers of the lanes are determined by an automatic thresholding The marker of the background is the complementary set of a dilation
APPLICATIONS (8)APPLICATIONS (8)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 32
bull Extraction of road marking by a top-hat transformbull Calculation of the distance function of the road marking between the markersbull watershed of the distance function
APPLICATIONS (9)APPLICATIONS (9)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 33
3D segmentations based on distance functions3D segmentations based on distance functions
Polyester foamPolyester foam
Distance functionDistance function
3D watershed3D watershed
APPLICATIONS (10)APPLICATIONS (10)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 34
3D segmentations based on gradients3D segmentations based on gradients
3D brain NMR image3D brain NMR image
APPLICATIONS (11)APPLICATIONS (11)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 35
CRITERIA BASED ON NUMERICAL RESIDUESCRITERIA BASED ON NUMERICAL RESIDUES
Starting Starting fromfrom twotwo sequencessequences of transformationsof transformations
( )iiIi
Sup ζψθ minus=isin
( ) 1maxarg +minus= iiq ζψ
ii ζψ geiψ andand
withwith wewe definedefine twotwo operatorsoperators
bullbull The The residualresidual TransformationTransformation
bullbull Its Its associatedassociated functionfunction
iζ
1+==
ii
ii
γζγψ
1+==
ii
ii
εζεψ
q q isis calledcalled quasiquasi-- distancedistance
θ θ isis namednamed ultimateultimate openingopeningq q isis the the granulometricgranulometric functionfunction
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 36
AtAt everyevery point x q(x) point x q(x) isis equalequal(up to the unit value) to the (up to the unit value) to the radius of the radius of the greatestgreatestsignificantsignificant cylindercylinder coveringcovering xx
HeapHeap of rocksof rocks UltimateUltimate openingopening
GranulometricGranulometric functionfunction
ULTIMATE OPENING ULTIMATE OPENING GRANULOMETRIC FUNCTIONGRANULOMETRIC FUNCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 37
EachEach thresholdthreshold λ λ of the of the granulometricgranulometric functionfunction q q isiserodederoded by a by a diskdisk of size kof size kλ λ (klt1)(klt1)
This This operationoperation producesproducesmarkers of blocks markers of blocks whosewhose size size isisproportionalproportional to the size of the to the size of the blockblock
MARKERS GENERATIONMARKERS GENERATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 38
ROCKS SEGMENTATIONROCKS SEGMENTATION
Markers Markers extractedextracted fromfrom the the granulometricgranulometric functionfunction provideprovidevaluablevaluable seedsseeds for a markerfor a marker--controlledcontrolled watershedwatershedsegmentation (size and segmentation (size and contrastcontrast criteriacriteria cancan bebe mixed)mixed)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 39
QUASIQUASI--DISTANCESDISTANCES
A A correctedcorrected quasiquasi--distance distance computedcomputed on a on a greytonegreytone image image providesprovides the the sizessizes of the flat (of the flat (homogeneoushomogeneous) ) regionsregions Markers Markers for a segmentation for a segmentation basedbased on size and on size and geometrygeometry
bull QuasiQuasi--distances distances performedperformed bothboth on on the image and the the image and the complementarycomplementary oneone
d dd drsquorsquobullbull Sup of the Sup of the resultsresults h=sup(ddh=sup(ddrsquorsquo))bullbull Markers extraction (maxima or Markers extraction (maxima or thresholdthreshold))bullbull WatershedWatershed of hof h
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 40
SEGMENTATION WITH QUASISEGMENTATION WITH QUASI--DISTANCESDISTANCES
cupcup
WS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 41
bullbull VideoVideo surveillance surveillance scenescene ((leftleft))bullbull QuasiQuasi--distance (distance (upperupper right)right)bullbull QuasiQuasi--distance of the distance of the invertedinvertedimage (image (lowerlower right) right)
ANOTHER EXAMPLEANOTHER EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 42
bullbull Markers of Markers of regionsregions ((leftleft))bullbull WatershedWatershed of the sup of the of the sup of the quasiquasi--distances (distances (upperupper right)right)bullbull The The movingmoving regionsregions are are detecteddetected((lowerlower right)right)
ANOTHER EXAMPLE (2)ANOTHER EXAMPLE (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 43
GRADIENT AND QUASIGRADIENT AND QUASI--DISTANCEDISTANCE
QuasiQuasi--distance distance cancan bebe computedcomputed on the on the invertedinverted gradient gradient functionfunctionbullbull OnlyOnly one quasione quasi--distance distance isis calculatedcalculatedbullbull HierarchyHierarchy of of regionsregions basedbased on on theirtheir relative relative contrastcontrast ((similarsimilar to the to the waterfallwaterfall))bullbull The The shapeshape of of regionsregions isis takentaken intointo accountaccount ((closureclosure of of imperfectlyimperfectly closedclosed
regionsregions))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 44
Cleavage fractures in SEM steel imagesCleavage fractures in SEM steel images
Distance function
Contrast function
First watershed
Second watershed
Common dams in both watersheds
DETAILED APPLICATIONSDETAILED APPLICATIONS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 45
Stereoscopic pairMarkers of the first image
Azimuth of distance function
Azimuth (2nd image)
The markers of the first image are thrown on the second onehellipand migrate along the steepest slope to give the new markers (in green)
DETAILED APPLICATIONS (2)DETAILED APPLICATIONS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 46
On the right contour of facets on the On the right contour of facets on the first image and the homologous ones first image and the homologous ones in the second imagein the second imageBelow displacement of a single facet Below displacement of a single facet which can be measured allowing the which can be measured allowing the computation of its altitudecomputation of its altitude
DETAILED APPLICATIONS (3)DETAILED APPLICATIONS (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 47
The PROMETHEUS projectThe PROMETHEUS project
Road segmentation and Road segmentation and obstacle detectionobstacle detection
Two phases
bull primary road or lane segmentation (hierarchical watershed) No information is shared between pictures in the sequencebull Definition of a roadlane model (sometimes very basic) and use of this model to build the markers which will be used for the segmentation of the next picture
DETAILED APPLICATIONS (4)DETAILED APPLICATIONS (4)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 48
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Initial imageInitial image
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 49
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
First segmentationFirst segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 50
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Second level of hierarchy Second level of hierarchy and marker extractionand marker extraction
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 51
Image iImage i
Lane segmentationLane segmentation(hierarchy)(hierarchy)
i = i+1 i = i+1
Phase I Phase I
Lane SegmentationLane Segmentation(hierarchy)(hierarchy)
Final segmentationFinal segmentation
DETAILED APPLICATIONS (5)DETAILED APPLICATIONS (5)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 52
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Sequential image at time iSequential image at time i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 53
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
MarkerMarker--controlled controlled segmentation of the lane segmentation of the lane (marker generated by the (marker generated by the previous image)previous image)
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 54
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
Those pixels belonging to the Those pixels belonging to the contours of the lane are contours of the lane are selectedselected
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 55
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculation
helliphellipand used to adjust a and used to adjust a laneroad modellaneroad model
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 56
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
The roadlane model leads to The roadlane model leads to the generation of a new the generation of a new markermarker
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 57
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 58
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11
The previous marker is used The previous marker is used to segment the current imageto segment the current image
Road modelRoad modelcalculationcalculation
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 59
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
And a new adjustment of the And a new adjustment of the roadlane model is performedroadlane model is performed
Note that despite its apparent complexity this phase is faster than phase I (no hierarchical segmentation)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 60
Phase IIPhase II
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Road modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
i = i+1i = i+1
Phase IPhase I
OKOK
Phase IPhase I
yesyes
nono
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
Image iImage i
Road modelRoad modelcalculationcalculationRoad modelRoad modelcalculationcalculation
MarkerMarkerimage iimage i--11
OKOK i = i+1i = i+1
Image iImage i
OKOK i = i+1i = i+1
Image iImage i
MarkerMarker--controlledcontrolledlane segmentationlane segmentation
MarkerMarkerimage iimage i--11Road modelRoad model
calculationcalculation
Demonstration of the process Demonstration of the process on a complete sequence (three on a complete sequence (three lanes road model)lanes road model)
DETAILED APPLICATIONS (6)DETAILED APPLICATIONS (6)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 61
It is not always possible to prevent overIt is not always possible to prevent over--segmentation by segmentation by markermarker--controlled watershed because it is not always possible controlled watershed because it is not always possible to find good markers andor segmentation criteriato find good markers andor segmentation criteria
ThereforeTherefore anotheranother approachesapproaches of the segmentation of the segmentation whichwhichare not are not basedbased on the a priori on the a priori selectionselection of markers of markers maymay bebeusefuluseful
DifferentDifferent algorithmsalgorithms existexist TheyThey aimaim atat definingdefining a a hierarchyhierarchyof segmentationsof segmentationsbullbull HierarchyHierarchy basedbased on extinction valueson extinction valuesbullbull HierarchyHierarchy basedbased on on waterfallswaterfallsbullbull HierarchyHierarchy basedbased on on pilingspilings
HIERARCHICAL SEGMENTATION WATERFALLSHIERARCHICAL SEGMENTATION WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 62
Valued watershed
The watershed of a function g is a set W(g)The valued watershed is the functionw(g) defined on W and equal to g on each point of W
WW
w(g)w(g)
Initial imageInitial image Gradient gGradient g
VALUED WATERSHEDVALUED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 63
Building the mosaic imageBuilding the mosaic image
bullbull Watershed of gradientWatershed of gradientbullbull For each minimum of For each minimum of gradient compute the gradient compute the corresponding grey valuecorresponding grey valuebullbull Fill in the catchment Fill in the catchment basin with this grey valuebasin with this grey value
MOSAIC IMAGE AND ITS GRADIENTMOSAIC IMAGE AND ITS GRADIENT
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 64
A simple illustration using a mosaic imageA simple illustration using a mosaic image
Despite the fact that the image is overDespite the fact that the image is over--segmented the white blob can be easily segmented the white blob can be easily distinguished from the background because at the same time thedistinguished from the background because at the same time the boundaries boundaries between the regions inside the blobs and the boundaries inside tbetween the regions inside the blobs and the boundaries inside the he background are less contrasted than the boundaries background are less contrasted than the boundaries wichwich separate the blob separate the blob and the background Both the blob and the background are and the background Both the blob and the background are markedmarked by by boundaries with a boundaries with a minimal contrastminimal contrast
OVEROVER--SEGMENTATION AND PERCEPTION SEGMENTATION AND PERCEPTION OF IMAGESOF IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 65
Arcs of minimal heightArcs of minimal heightIn the mosaic image each arc In the mosaic image each arc ccijijseparates two catchment basins separates two catchment basins CBCBiiand and CBCBjj The valuation The valuation vvijij of the arc of the arc is given byis given by
vvijij = | = | ggii -- ggjj ||
where where ggii and and ggjj are the grey values are the grey values in the catchment basinsin the catchment basinsAn arc An arc ccijij is said to be minimal if its is said to be minimal if its valuation is lower than those of all valuation is lower than those of all the other arcs surrounding the other arcs surrounding CBCBiiand and CBCBjj
GRAPH DEFINITIONGRAPH DEFINITION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 66
Definition of a new graphDefinition of a new graph
bullbull its vertices correspond to the arcs of the its vertices correspond to the arcs of the gradient mosaicgradient mosaicbullbull its edges link all arcs surrounding the same its edges link all arcs surrounding the same catchment basincatchment basinbullbull each vertex is valued by the arc valuation as each vertex is valued by the arc valuation as defined in the gradient mosaic defined in the gradient mosaic
In this representation the arcs surrounding In this representation the arcs surrounding the same catchment basin are adjacent the same catchment basin are adjacent Therefore minimal arcs can be connected Therefore minimal arcs can be connected although it is not the case in the gradient although it is not the case in the gradient mosaic as illustrated above (yellow summits mosaic as illustrated above (yellow summits correspond to minimal arcs)correspond to minimal arcs)
GRAPH DEFINITION AND GRAPH DEFINITION AND ASSOCIATED WATERSHEDASSOCIATED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 67
Flooding step 1 (in blue)
The most significant contours of the The most significant contours of the mosaic image correspond to those mosaic image correspond to those separating regions marked by separating regions marked by minimal arcs They are the watershed minimal arcs They are the watershed lines of the watershed transform lines of the watershed transform defined on the previously defined defined on the previously defined graphgraph
Step 2 two CBs in blue amp green Step 3 first dams in red
GRAPH DEFINITION AND ASSOCIATED GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)WATERSHED (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 68
Step 4 Final watershed
Arcs of the gradient mosaic corresponding to the watershed lines
GRAPH DEFINITION AND ASSOCIATED GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)WATERSHED (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 69
The previously defined graph is a 3D The previously defined graph is a 3D valued graph which is not very handyvalued graph which is not very handy
This graph can be transformed into a This graph can be transformed into a planar one by the following procedureplanar one by the following procedurebullbull A new vertex is added in each A new vertex is added in each catchment basincatchment basinbullbull The previous edges are replaced by two The previous edges are replaced by two successive edges linking the original successive edges linking the original vertices through the new onevertices through the new onebull The valuation of the new vertex is given byThe valuation of the new vertex is given by
min (min (vvijij))where where vvijij are the valuations of the arcs surrounding the catchment basinare the valuations of the arcs surrounding the catchment basinij
FROM A 3D to A PLANAR GRAPHFROM A 3D to A PLANAR GRAPH
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 70
The hierarchical imageThe hierarchical image
mosaic
gradient mosaic
hierarchical image
An image named hierarchical image can be An image named hierarchical image can be build from the planar graph The catchment build from the planar graph The catchment basins of the gradient mosaic are filled with basins of the gradient mosaic are filled with grey values corresponding to the valuation of grey values corresponding to the valuation of the new added verticesthe new added verticesThe watersheds of this hierarchical image The watersheds of this hierarchical image give the higher level of hierarchy (with some give the higher level of hierarchy (with some restrictions)restrictions)
IMAGE REPRESENTATIONIMAGE REPRESENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 71
AlsoAlso calledcalled improperlyimproperly saddlesaddle zoneszones(the FOZ notion has (the FOZ notion has nothingnothing to do to do withwith the the classicalclassical saddlesaddle zones Itzones Itrsquorsquos not a s not a local notion and as the WTS local notion and as the WTS therethere isis no no wayway to know a priori if a to know a priori if a givengiven point point of the image of the image belongsbelongs or not to a FOZ)or not to a FOZ)
LowerLower catchmentcatchment basinbasinIt It isis the part of the the part of the catchmentcatchment basin basin whichwhich isis floodedflooded beforebefore the first the first overflowoverflow((whichwhich occursoccurs throughthrough the the lowestlowest FOZ)FOZ)
First overflows
FIRST OVERFLOW ZONES (FOZ)FIRST OVERFLOW ZONES (FOZ)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 72
IntroductionIntroductionConsider the function f and its Consider the function f and its watershed Various catchment watershed Various catchment basins are numbered from 1 to 9 basins are numbered from 1 to 9 Consider the flooding from the Consider the flooding from the minimum m minimum m
When filling CB1 an overflow When filling CB1 an overflow occurs towards CB2occurs towards CB2
Now if we fill in CB2 the first Now if we fill in CB2 the first overflow occurs towards CB1overflow occurs towards CB1
In this case overflows In this case overflows (waterfalls) are symmetrical(waterfalls) are symmetrical
Therefore the part of the Therefore the part of the watershed line separating CB1 watershed line separating CB1 from CB2 can be removed and from CB2 can be removed and the floods in CB1 and CB2 can the floods in CB1 and CB2 can be mergedbe merged
1
WATERFALLSWATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 73
If this flooding process is iterated If this flooding process is iterated the flood invades CB3 which in the flood invades CB3 which in return when flooded pours into the return when flooded pours into the merged basins CB1 and CB2 Here merged basins CB1 and CB2 Here again the waterfalls being again the waterfalls being symmetrical CB3 is merged to the symmetrical CB3 is merged to the floodflood
Step by step because in each case Step by step because in each case waterfalls are symmetrical all the waterfalls are symmetrical all the catchment basins from 1 to 6 are catchment basins from 1 to 6 are mergedmerged
But when the flood pours into CB7 But when the flood pours into CB7 the situation changes Now if we the situation changes Now if we flood CB7 the waterfall is no longer flood CB7 the waterfall is no longer symmetrical Therefore the symmetrical Therefore the watershed line between CB7 and the watershed line between CB7 and the merged basins must be preservedmerged basins must be preserved
WATERFALLS (2)WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 74
Floodingfrom here
The previous process does not The previous process does not work if we start from any work if we start from any basin However the flooding in basin However the flooding in the end reaches the significant the end reaches the significant CBsCBs
The successive floods generate the lower The successive floods generate the lower catchment basins associated with each CB catchment basins associated with each CB (flood just before the overflow through the (flood just before the overflow through the lower saddle zone)lower saddle zone)
This can be achieved directly by a dual This can be achieved directly by a dual reconstruction of the initial function by the reconstruction of the initial function by the lower saddle zones lower saddle zones
SIGNIFICANT CBARCS AND RECONSTRUCTIONSIGNIFICANT CBARCS AND RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 75
InsteadInstead of of usingusing FOZ (not FOZ (not easyeasy to to detectdetect themthem) the ) the wholewhole set of set of watershedwatershed lineslines maymay bebe usedused The The resultresult willwill bebe the the samesame because the because the FOZ FOZ isis the the regionregion atat the the lowestlowest altitude altitude borderingbordering the the catchmentcatchmentbasinbasin
bullbull f initial f initial functionfunctionbullbull let us let us definedefine ggg (x) = f (x) if and g (x) = f (x) if and onlyonly if x if x belongsbelongs to the to the watershedswatersheds of fof fg (x) = max if notg (x) = max if notbullbull h = Rh = Rf f (g) (g) resultresult of the dual of the dual reconstruction of f by g reconstruction of f by g isis alsoalso calledcalledhierarchicalhierarchical imageimagebullbull W(h) W(h) watershedwatershed of h of h producesproduces the the hierarchicalhierarchical segmentation of segmentation of higherhigher levellevel
WhenWhen f f isis a a valuedvalued WTS WTS thisthis hierarchicalhierarchical image image isis identicalidentical to to the the previousprevious definitiondefinition
Lower catchmentbasins
RECONSTRUCTION AND HIERARCHICAL IMAGERECONSTRUCTION AND HIERARCHICAL IMAGE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 76
In this case the hierarchical approach and the waterfall approaIn this case the hierarchical approach and the waterfall approach ch are identical The waterfall transformation is the generalisatioare identical The waterfall transformation is the generalisation for n for any function of the hierarchical approachany function of the hierarchical approach
Gradient mosaic image
Hierarchical image Waterfall image
The minimal valuation of the catchment The minimal valuation of the catchment basin corresponds to the height of the basin corresponds to the height of the lower FOZ This valuation produces the lower FOZ This valuation produces the same result as the reconstruction of the same result as the reconstruction of the gradient mosaic function by the lower gradient mosaic function by the lower FOZFOZ
WATERFALLS AND MOSAIC IMAGESWATERFALLS AND MOSAIC IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 62
Valued watershed
The watershed of a function g is a set W(g)The valued watershed is the functionw(g) defined on W and equal to g on each point of W
WW
w(g)w(g)
Initial imageInitial image Gradient gGradient g
VALUED WATERSHEDVALUED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 63
Building the mosaic imageBuilding the mosaic image
bullbull Watershed of gradientWatershed of gradientbullbull For each minimum of For each minimum of gradient compute the gradient compute the corresponding grey valuecorresponding grey valuebullbull Fill in the catchment Fill in the catchment basin with this grey valuebasin with this grey value
MOSAIC IMAGE AND ITS GRADIENTMOSAIC IMAGE AND ITS GRADIENT
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 64
A simple illustration using a mosaic imageA simple illustration using a mosaic image
Despite the fact that the image is overDespite the fact that the image is over--segmented the white blob can be easily segmented the white blob can be easily distinguished from the background because at the same time thedistinguished from the background because at the same time the boundaries boundaries between the regions inside the blobs and the boundaries inside tbetween the regions inside the blobs and the boundaries inside the he background are less contrasted than the boundaries background are less contrasted than the boundaries wichwich separate the blob separate the blob and the background Both the blob and the background are and the background Both the blob and the background are markedmarked by by boundaries with a boundaries with a minimal contrastminimal contrast
OVEROVER--SEGMENTATION AND PERCEPTION SEGMENTATION AND PERCEPTION OF IMAGESOF IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 65
Arcs of minimal heightArcs of minimal heightIn the mosaic image each arc In the mosaic image each arc ccijijseparates two catchment basins separates two catchment basins CBCBiiand and CBCBjj The valuation The valuation vvijij of the arc of the arc is given byis given by
vvijij = | = | ggii -- ggjj ||
where where ggii and and ggjj are the grey values are the grey values in the catchment basinsin the catchment basinsAn arc An arc ccijij is said to be minimal if its is said to be minimal if its valuation is lower than those of all valuation is lower than those of all the other arcs surrounding the other arcs surrounding CBCBiiand and CBCBjj
GRAPH DEFINITIONGRAPH DEFINITION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 66
Definition of a new graphDefinition of a new graph
bullbull its vertices correspond to the arcs of the its vertices correspond to the arcs of the gradient mosaicgradient mosaicbullbull its edges link all arcs surrounding the same its edges link all arcs surrounding the same catchment basincatchment basinbullbull each vertex is valued by the arc valuation as each vertex is valued by the arc valuation as defined in the gradient mosaic defined in the gradient mosaic
In this representation the arcs surrounding In this representation the arcs surrounding the same catchment basin are adjacent the same catchment basin are adjacent Therefore minimal arcs can be connected Therefore minimal arcs can be connected although it is not the case in the gradient although it is not the case in the gradient mosaic as illustrated above (yellow summits mosaic as illustrated above (yellow summits correspond to minimal arcs)correspond to minimal arcs)
GRAPH DEFINITION AND GRAPH DEFINITION AND ASSOCIATED WATERSHEDASSOCIATED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 67
Flooding step 1 (in blue)
The most significant contours of the The most significant contours of the mosaic image correspond to those mosaic image correspond to those separating regions marked by separating regions marked by minimal arcs They are the watershed minimal arcs They are the watershed lines of the watershed transform lines of the watershed transform defined on the previously defined defined on the previously defined graphgraph
Step 2 two CBs in blue amp green Step 3 first dams in red
GRAPH DEFINITION AND ASSOCIATED GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)WATERSHED (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 68
Step 4 Final watershed
Arcs of the gradient mosaic corresponding to the watershed lines
GRAPH DEFINITION AND ASSOCIATED GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)WATERSHED (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 69
The previously defined graph is a 3D The previously defined graph is a 3D valued graph which is not very handyvalued graph which is not very handy
This graph can be transformed into a This graph can be transformed into a planar one by the following procedureplanar one by the following procedurebullbull A new vertex is added in each A new vertex is added in each catchment basincatchment basinbullbull The previous edges are replaced by two The previous edges are replaced by two successive edges linking the original successive edges linking the original vertices through the new onevertices through the new onebull The valuation of the new vertex is given byThe valuation of the new vertex is given by
min (min (vvijij))where where vvijij are the valuations of the arcs surrounding the catchment basinare the valuations of the arcs surrounding the catchment basinij
FROM A 3D to A PLANAR GRAPHFROM A 3D to A PLANAR GRAPH
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 70
The hierarchical imageThe hierarchical image
mosaic
gradient mosaic
hierarchical image
An image named hierarchical image can be An image named hierarchical image can be build from the planar graph The catchment build from the planar graph The catchment basins of the gradient mosaic are filled with basins of the gradient mosaic are filled with grey values corresponding to the valuation of grey values corresponding to the valuation of the new added verticesthe new added verticesThe watersheds of this hierarchical image The watersheds of this hierarchical image give the higher level of hierarchy (with some give the higher level of hierarchy (with some restrictions)restrictions)
IMAGE REPRESENTATIONIMAGE REPRESENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 71
AlsoAlso calledcalled improperlyimproperly saddlesaddle zoneszones(the FOZ notion has (the FOZ notion has nothingnothing to do to do withwith the the classicalclassical saddlesaddle zones Itzones Itrsquorsquos not a s not a local notion and as the WTS local notion and as the WTS therethere isis no no wayway to know a priori if a to know a priori if a givengiven point point of the image of the image belongsbelongs or not to a FOZ)or not to a FOZ)
LowerLower catchmentcatchment basinbasinIt It isis the part of the the part of the catchmentcatchment basin basin whichwhich isis floodedflooded beforebefore the first the first overflowoverflow((whichwhich occursoccurs throughthrough the the lowestlowest FOZ)FOZ)
First overflows
FIRST OVERFLOW ZONES (FOZ)FIRST OVERFLOW ZONES (FOZ)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 72
IntroductionIntroductionConsider the function f and its Consider the function f and its watershed Various catchment watershed Various catchment basins are numbered from 1 to 9 basins are numbered from 1 to 9 Consider the flooding from the Consider the flooding from the minimum m minimum m
When filling CB1 an overflow When filling CB1 an overflow occurs towards CB2occurs towards CB2
Now if we fill in CB2 the first Now if we fill in CB2 the first overflow occurs towards CB1overflow occurs towards CB1
In this case overflows In this case overflows (waterfalls) are symmetrical(waterfalls) are symmetrical
Therefore the part of the Therefore the part of the watershed line separating CB1 watershed line separating CB1 from CB2 can be removed and from CB2 can be removed and the floods in CB1 and CB2 can the floods in CB1 and CB2 can be mergedbe merged
1
WATERFALLSWATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 73
If this flooding process is iterated If this flooding process is iterated the flood invades CB3 which in the flood invades CB3 which in return when flooded pours into the return when flooded pours into the merged basins CB1 and CB2 Here merged basins CB1 and CB2 Here again the waterfalls being again the waterfalls being symmetrical CB3 is merged to the symmetrical CB3 is merged to the floodflood
Step by step because in each case Step by step because in each case waterfalls are symmetrical all the waterfalls are symmetrical all the catchment basins from 1 to 6 are catchment basins from 1 to 6 are mergedmerged
But when the flood pours into CB7 But when the flood pours into CB7 the situation changes Now if we the situation changes Now if we flood CB7 the waterfall is no longer flood CB7 the waterfall is no longer symmetrical Therefore the symmetrical Therefore the watershed line between CB7 and the watershed line between CB7 and the merged basins must be preservedmerged basins must be preserved
WATERFALLS (2)WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 74
Floodingfrom here
The previous process does not The previous process does not work if we start from any work if we start from any basin However the flooding in basin However the flooding in the end reaches the significant the end reaches the significant CBsCBs
The successive floods generate the lower The successive floods generate the lower catchment basins associated with each CB catchment basins associated with each CB (flood just before the overflow through the (flood just before the overflow through the lower saddle zone)lower saddle zone)
This can be achieved directly by a dual This can be achieved directly by a dual reconstruction of the initial function by the reconstruction of the initial function by the lower saddle zones lower saddle zones
SIGNIFICANT CBARCS AND RECONSTRUCTIONSIGNIFICANT CBARCS AND RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 75
InsteadInstead of of usingusing FOZ (not FOZ (not easyeasy to to detectdetect themthem) the ) the wholewhole set of set of watershedwatershed lineslines maymay bebe usedused The The resultresult willwill bebe the the samesame because the because the FOZ FOZ isis the the regionregion atat the the lowestlowest altitude altitude borderingbordering the the catchmentcatchmentbasinbasin
bullbull f initial f initial functionfunctionbullbull let us let us definedefine ggg (x) = f (x) if and g (x) = f (x) if and onlyonly if x if x belongsbelongs to the to the watershedswatersheds of fof fg (x) = max if notg (x) = max if notbullbull h = Rh = Rf f (g) (g) resultresult of the dual of the dual reconstruction of f by g reconstruction of f by g isis alsoalso calledcalledhierarchicalhierarchical imageimagebullbull W(h) W(h) watershedwatershed of h of h producesproduces the the hierarchicalhierarchical segmentation of segmentation of higherhigher levellevel
WhenWhen f f isis a a valuedvalued WTS WTS thisthis hierarchicalhierarchical image image isis identicalidentical to to the the previousprevious definitiondefinition
Lower catchmentbasins
RECONSTRUCTION AND HIERARCHICAL IMAGERECONSTRUCTION AND HIERARCHICAL IMAGE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 76
In this case the hierarchical approach and the waterfall approaIn this case the hierarchical approach and the waterfall approach ch are identical The waterfall transformation is the generalisatioare identical The waterfall transformation is the generalisation for n for any function of the hierarchical approachany function of the hierarchical approach
Gradient mosaic image
Hierarchical image Waterfall image
The minimal valuation of the catchment The minimal valuation of the catchment basin corresponds to the height of the basin corresponds to the height of the lower FOZ This valuation produces the lower FOZ This valuation produces the same result as the reconstruction of the same result as the reconstruction of the gradient mosaic function by the lower gradient mosaic function by the lower FOZFOZ
WATERFALLS AND MOSAIC IMAGESWATERFALLS AND MOSAIC IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 63
Building the mosaic imageBuilding the mosaic image
bullbull Watershed of gradientWatershed of gradientbullbull For each minimum of For each minimum of gradient compute the gradient compute the corresponding grey valuecorresponding grey valuebullbull Fill in the catchment Fill in the catchment basin with this grey valuebasin with this grey value
MOSAIC IMAGE AND ITS GRADIENTMOSAIC IMAGE AND ITS GRADIENT
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 64
A simple illustration using a mosaic imageA simple illustration using a mosaic image
Despite the fact that the image is overDespite the fact that the image is over--segmented the white blob can be easily segmented the white blob can be easily distinguished from the background because at the same time thedistinguished from the background because at the same time the boundaries boundaries between the regions inside the blobs and the boundaries inside tbetween the regions inside the blobs and the boundaries inside the he background are less contrasted than the boundaries background are less contrasted than the boundaries wichwich separate the blob separate the blob and the background Both the blob and the background are and the background Both the blob and the background are markedmarked by by boundaries with a boundaries with a minimal contrastminimal contrast
OVEROVER--SEGMENTATION AND PERCEPTION SEGMENTATION AND PERCEPTION OF IMAGESOF IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 65
Arcs of minimal heightArcs of minimal heightIn the mosaic image each arc In the mosaic image each arc ccijijseparates two catchment basins separates two catchment basins CBCBiiand and CBCBjj The valuation The valuation vvijij of the arc of the arc is given byis given by
vvijij = | = | ggii -- ggjj ||
where where ggii and and ggjj are the grey values are the grey values in the catchment basinsin the catchment basinsAn arc An arc ccijij is said to be minimal if its is said to be minimal if its valuation is lower than those of all valuation is lower than those of all the other arcs surrounding the other arcs surrounding CBCBiiand and CBCBjj
GRAPH DEFINITIONGRAPH DEFINITION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 66
Definition of a new graphDefinition of a new graph
bullbull its vertices correspond to the arcs of the its vertices correspond to the arcs of the gradient mosaicgradient mosaicbullbull its edges link all arcs surrounding the same its edges link all arcs surrounding the same catchment basincatchment basinbullbull each vertex is valued by the arc valuation as each vertex is valued by the arc valuation as defined in the gradient mosaic defined in the gradient mosaic
In this representation the arcs surrounding In this representation the arcs surrounding the same catchment basin are adjacent the same catchment basin are adjacent Therefore minimal arcs can be connected Therefore minimal arcs can be connected although it is not the case in the gradient although it is not the case in the gradient mosaic as illustrated above (yellow summits mosaic as illustrated above (yellow summits correspond to minimal arcs)correspond to minimal arcs)
GRAPH DEFINITION AND GRAPH DEFINITION AND ASSOCIATED WATERSHEDASSOCIATED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 67
Flooding step 1 (in blue)
The most significant contours of the The most significant contours of the mosaic image correspond to those mosaic image correspond to those separating regions marked by separating regions marked by minimal arcs They are the watershed minimal arcs They are the watershed lines of the watershed transform lines of the watershed transform defined on the previously defined defined on the previously defined graphgraph
Step 2 two CBs in blue amp green Step 3 first dams in red
GRAPH DEFINITION AND ASSOCIATED GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)WATERSHED (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 68
Step 4 Final watershed
Arcs of the gradient mosaic corresponding to the watershed lines
GRAPH DEFINITION AND ASSOCIATED GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)WATERSHED (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 69
The previously defined graph is a 3D The previously defined graph is a 3D valued graph which is not very handyvalued graph which is not very handy
This graph can be transformed into a This graph can be transformed into a planar one by the following procedureplanar one by the following procedurebullbull A new vertex is added in each A new vertex is added in each catchment basincatchment basinbullbull The previous edges are replaced by two The previous edges are replaced by two successive edges linking the original successive edges linking the original vertices through the new onevertices through the new onebull The valuation of the new vertex is given byThe valuation of the new vertex is given by
min (min (vvijij))where where vvijij are the valuations of the arcs surrounding the catchment basinare the valuations of the arcs surrounding the catchment basinij
FROM A 3D to A PLANAR GRAPHFROM A 3D to A PLANAR GRAPH
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 70
The hierarchical imageThe hierarchical image
mosaic
gradient mosaic
hierarchical image
An image named hierarchical image can be An image named hierarchical image can be build from the planar graph The catchment build from the planar graph The catchment basins of the gradient mosaic are filled with basins of the gradient mosaic are filled with grey values corresponding to the valuation of grey values corresponding to the valuation of the new added verticesthe new added verticesThe watersheds of this hierarchical image The watersheds of this hierarchical image give the higher level of hierarchy (with some give the higher level of hierarchy (with some restrictions)restrictions)
IMAGE REPRESENTATIONIMAGE REPRESENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 71
AlsoAlso calledcalled improperlyimproperly saddlesaddle zoneszones(the FOZ notion has (the FOZ notion has nothingnothing to do to do withwith the the classicalclassical saddlesaddle zones Itzones Itrsquorsquos not a s not a local notion and as the WTS local notion and as the WTS therethere isis no no wayway to know a priori if a to know a priori if a givengiven point point of the image of the image belongsbelongs or not to a FOZ)or not to a FOZ)
LowerLower catchmentcatchment basinbasinIt It isis the part of the the part of the catchmentcatchment basin basin whichwhich isis floodedflooded beforebefore the first the first overflowoverflow((whichwhich occursoccurs throughthrough the the lowestlowest FOZ)FOZ)
First overflows
FIRST OVERFLOW ZONES (FOZ)FIRST OVERFLOW ZONES (FOZ)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 72
IntroductionIntroductionConsider the function f and its Consider the function f and its watershed Various catchment watershed Various catchment basins are numbered from 1 to 9 basins are numbered from 1 to 9 Consider the flooding from the Consider the flooding from the minimum m minimum m
When filling CB1 an overflow When filling CB1 an overflow occurs towards CB2occurs towards CB2
Now if we fill in CB2 the first Now if we fill in CB2 the first overflow occurs towards CB1overflow occurs towards CB1
In this case overflows In this case overflows (waterfalls) are symmetrical(waterfalls) are symmetrical
Therefore the part of the Therefore the part of the watershed line separating CB1 watershed line separating CB1 from CB2 can be removed and from CB2 can be removed and the floods in CB1 and CB2 can the floods in CB1 and CB2 can be mergedbe merged
1
WATERFALLSWATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 73
If this flooding process is iterated If this flooding process is iterated the flood invades CB3 which in the flood invades CB3 which in return when flooded pours into the return when flooded pours into the merged basins CB1 and CB2 Here merged basins CB1 and CB2 Here again the waterfalls being again the waterfalls being symmetrical CB3 is merged to the symmetrical CB3 is merged to the floodflood
Step by step because in each case Step by step because in each case waterfalls are symmetrical all the waterfalls are symmetrical all the catchment basins from 1 to 6 are catchment basins from 1 to 6 are mergedmerged
But when the flood pours into CB7 But when the flood pours into CB7 the situation changes Now if we the situation changes Now if we flood CB7 the waterfall is no longer flood CB7 the waterfall is no longer symmetrical Therefore the symmetrical Therefore the watershed line between CB7 and the watershed line between CB7 and the merged basins must be preservedmerged basins must be preserved
WATERFALLS (2)WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 74
Floodingfrom here
The previous process does not The previous process does not work if we start from any work if we start from any basin However the flooding in basin However the flooding in the end reaches the significant the end reaches the significant CBsCBs
The successive floods generate the lower The successive floods generate the lower catchment basins associated with each CB catchment basins associated with each CB (flood just before the overflow through the (flood just before the overflow through the lower saddle zone)lower saddle zone)
This can be achieved directly by a dual This can be achieved directly by a dual reconstruction of the initial function by the reconstruction of the initial function by the lower saddle zones lower saddle zones
SIGNIFICANT CBARCS AND RECONSTRUCTIONSIGNIFICANT CBARCS AND RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 75
InsteadInstead of of usingusing FOZ (not FOZ (not easyeasy to to detectdetect themthem) the ) the wholewhole set of set of watershedwatershed lineslines maymay bebe usedused The The resultresult willwill bebe the the samesame because the because the FOZ FOZ isis the the regionregion atat the the lowestlowest altitude altitude borderingbordering the the catchmentcatchmentbasinbasin
bullbull f initial f initial functionfunctionbullbull let us let us definedefine ggg (x) = f (x) if and g (x) = f (x) if and onlyonly if x if x belongsbelongs to the to the watershedswatersheds of fof fg (x) = max if notg (x) = max if notbullbull h = Rh = Rf f (g) (g) resultresult of the dual of the dual reconstruction of f by g reconstruction of f by g isis alsoalso calledcalledhierarchicalhierarchical imageimagebullbull W(h) W(h) watershedwatershed of h of h producesproduces the the hierarchicalhierarchical segmentation of segmentation of higherhigher levellevel
WhenWhen f f isis a a valuedvalued WTS WTS thisthis hierarchicalhierarchical image image isis identicalidentical to to the the previousprevious definitiondefinition
Lower catchmentbasins
RECONSTRUCTION AND HIERARCHICAL IMAGERECONSTRUCTION AND HIERARCHICAL IMAGE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 76
In this case the hierarchical approach and the waterfall approaIn this case the hierarchical approach and the waterfall approach ch are identical The waterfall transformation is the generalisatioare identical The waterfall transformation is the generalisation for n for any function of the hierarchical approachany function of the hierarchical approach
Gradient mosaic image
Hierarchical image Waterfall image
The minimal valuation of the catchment The minimal valuation of the catchment basin corresponds to the height of the basin corresponds to the height of the lower FOZ This valuation produces the lower FOZ This valuation produces the same result as the reconstruction of the same result as the reconstruction of the gradient mosaic function by the lower gradient mosaic function by the lower FOZFOZ
WATERFALLS AND MOSAIC IMAGESWATERFALLS AND MOSAIC IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 64
A simple illustration using a mosaic imageA simple illustration using a mosaic image
Despite the fact that the image is overDespite the fact that the image is over--segmented the white blob can be easily segmented the white blob can be easily distinguished from the background because at the same time thedistinguished from the background because at the same time the boundaries boundaries between the regions inside the blobs and the boundaries inside tbetween the regions inside the blobs and the boundaries inside the he background are less contrasted than the boundaries background are less contrasted than the boundaries wichwich separate the blob separate the blob and the background Both the blob and the background are and the background Both the blob and the background are markedmarked by by boundaries with a boundaries with a minimal contrastminimal contrast
OVEROVER--SEGMENTATION AND PERCEPTION SEGMENTATION AND PERCEPTION OF IMAGESOF IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 65
Arcs of minimal heightArcs of minimal heightIn the mosaic image each arc In the mosaic image each arc ccijijseparates two catchment basins separates two catchment basins CBCBiiand and CBCBjj The valuation The valuation vvijij of the arc of the arc is given byis given by
vvijij = | = | ggii -- ggjj ||
where where ggii and and ggjj are the grey values are the grey values in the catchment basinsin the catchment basinsAn arc An arc ccijij is said to be minimal if its is said to be minimal if its valuation is lower than those of all valuation is lower than those of all the other arcs surrounding the other arcs surrounding CBCBiiand and CBCBjj
GRAPH DEFINITIONGRAPH DEFINITION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 66
Definition of a new graphDefinition of a new graph
bullbull its vertices correspond to the arcs of the its vertices correspond to the arcs of the gradient mosaicgradient mosaicbullbull its edges link all arcs surrounding the same its edges link all arcs surrounding the same catchment basincatchment basinbullbull each vertex is valued by the arc valuation as each vertex is valued by the arc valuation as defined in the gradient mosaic defined in the gradient mosaic
In this representation the arcs surrounding In this representation the arcs surrounding the same catchment basin are adjacent the same catchment basin are adjacent Therefore minimal arcs can be connected Therefore minimal arcs can be connected although it is not the case in the gradient although it is not the case in the gradient mosaic as illustrated above (yellow summits mosaic as illustrated above (yellow summits correspond to minimal arcs)correspond to minimal arcs)
GRAPH DEFINITION AND GRAPH DEFINITION AND ASSOCIATED WATERSHEDASSOCIATED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 67
Flooding step 1 (in blue)
The most significant contours of the The most significant contours of the mosaic image correspond to those mosaic image correspond to those separating regions marked by separating regions marked by minimal arcs They are the watershed minimal arcs They are the watershed lines of the watershed transform lines of the watershed transform defined on the previously defined defined on the previously defined graphgraph
Step 2 two CBs in blue amp green Step 3 first dams in red
GRAPH DEFINITION AND ASSOCIATED GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)WATERSHED (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 68
Step 4 Final watershed
Arcs of the gradient mosaic corresponding to the watershed lines
GRAPH DEFINITION AND ASSOCIATED GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)WATERSHED (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 69
The previously defined graph is a 3D The previously defined graph is a 3D valued graph which is not very handyvalued graph which is not very handy
This graph can be transformed into a This graph can be transformed into a planar one by the following procedureplanar one by the following procedurebullbull A new vertex is added in each A new vertex is added in each catchment basincatchment basinbullbull The previous edges are replaced by two The previous edges are replaced by two successive edges linking the original successive edges linking the original vertices through the new onevertices through the new onebull The valuation of the new vertex is given byThe valuation of the new vertex is given by
min (min (vvijij))where where vvijij are the valuations of the arcs surrounding the catchment basinare the valuations of the arcs surrounding the catchment basinij
FROM A 3D to A PLANAR GRAPHFROM A 3D to A PLANAR GRAPH
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 70
The hierarchical imageThe hierarchical image
mosaic
gradient mosaic
hierarchical image
An image named hierarchical image can be An image named hierarchical image can be build from the planar graph The catchment build from the planar graph The catchment basins of the gradient mosaic are filled with basins of the gradient mosaic are filled with grey values corresponding to the valuation of grey values corresponding to the valuation of the new added verticesthe new added verticesThe watersheds of this hierarchical image The watersheds of this hierarchical image give the higher level of hierarchy (with some give the higher level of hierarchy (with some restrictions)restrictions)
IMAGE REPRESENTATIONIMAGE REPRESENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 71
AlsoAlso calledcalled improperlyimproperly saddlesaddle zoneszones(the FOZ notion has (the FOZ notion has nothingnothing to do to do withwith the the classicalclassical saddlesaddle zones Itzones Itrsquorsquos not a s not a local notion and as the WTS local notion and as the WTS therethere isis no no wayway to know a priori if a to know a priori if a givengiven point point of the image of the image belongsbelongs or not to a FOZ)or not to a FOZ)
LowerLower catchmentcatchment basinbasinIt It isis the part of the the part of the catchmentcatchment basin basin whichwhich isis floodedflooded beforebefore the first the first overflowoverflow((whichwhich occursoccurs throughthrough the the lowestlowest FOZ)FOZ)
First overflows
FIRST OVERFLOW ZONES (FOZ)FIRST OVERFLOW ZONES (FOZ)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 72
IntroductionIntroductionConsider the function f and its Consider the function f and its watershed Various catchment watershed Various catchment basins are numbered from 1 to 9 basins are numbered from 1 to 9 Consider the flooding from the Consider the flooding from the minimum m minimum m
When filling CB1 an overflow When filling CB1 an overflow occurs towards CB2occurs towards CB2
Now if we fill in CB2 the first Now if we fill in CB2 the first overflow occurs towards CB1overflow occurs towards CB1
In this case overflows In this case overflows (waterfalls) are symmetrical(waterfalls) are symmetrical
Therefore the part of the Therefore the part of the watershed line separating CB1 watershed line separating CB1 from CB2 can be removed and from CB2 can be removed and the floods in CB1 and CB2 can the floods in CB1 and CB2 can be mergedbe merged
1
WATERFALLSWATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 73
If this flooding process is iterated If this flooding process is iterated the flood invades CB3 which in the flood invades CB3 which in return when flooded pours into the return when flooded pours into the merged basins CB1 and CB2 Here merged basins CB1 and CB2 Here again the waterfalls being again the waterfalls being symmetrical CB3 is merged to the symmetrical CB3 is merged to the floodflood
Step by step because in each case Step by step because in each case waterfalls are symmetrical all the waterfalls are symmetrical all the catchment basins from 1 to 6 are catchment basins from 1 to 6 are mergedmerged
But when the flood pours into CB7 But when the flood pours into CB7 the situation changes Now if we the situation changes Now if we flood CB7 the waterfall is no longer flood CB7 the waterfall is no longer symmetrical Therefore the symmetrical Therefore the watershed line between CB7 and the watershed line between CB7 and the merged basins must be preservedmerged basins must be preserved
WATERFALLS (2)WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 74
Floodingfrom here
The previous process does not The previous process does not work if we start from any work if we start from any basin However the flooding in basin However the flooding in the end reaches the significant the end reaches the significant CBsCBs
The successive floods generate the lower The successive floods generate the lower catchment basins associated with each CB catchment basins associated with each CB (flood just before the overflow through the (flood just before the overflow through the lower saddle zone)lower saddle zone)
This can be achieved directly by a dual This can be achieved directly by a dual reconstruction of the initial function by the reconstruction of the initial function by the lower saddle zones lower saddle zones
SIGNIFICANT CBARCS AND RECONSTRUCTIONSIGNIFICANT CBARCS AND RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 75
InsteadInstead of of usingusing FOZ (not FOZ (not easyeasy to to detectdetect themthem) the ) the wholewhole set of set of watershedwatershed lineslines maymay bebe usedused The The resultresult willwill bebe the the samesame because the because the FOZ FOZ isis the the regionregion atat the the lowestlowest altitude altitude borderingbordering the the catchmentcatchmentbasinbasin
bullbull f initial f initial functionfunctionbullbull let us let us definedefine ggg (x) = f (x) if and g (x) = f (x) if and onlyonly if x if x belongsbelongs to the to the watershedswatersheds of fof fg (x) = max if notg (x) = max if notbullbull h = Rh = Rf f (g) (g) resultresult of the dual of the dual reconstruction of f by g reconstruction of f by g isis alsoalso calledcalledhierarchicalhierarchical imageimagebullbull W(h) W(h) watershedwatershed of h of h producesproduces the the hierarchicalhierarchical segmentation of segmentation of higherhigher levellevel
WhenWhen f f isis a a valuedvalued WTS WTS thisthis hierarchicalhierarchical image image isis identicalidentical to to the the previousprevious definitiondefinition
Lower catchmentbasins
RECONSTRUCTION AND HIERARCHICAL IMAGERECONSTRUCTION AND HIERARCHICAL IMAGE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 76
In this case the hierarchical approach and the waterfall approaIn this case the hierarchical approach and the waterfall approach ch are identical The waterfall transformation is the generalisatioare identical The waterfall transformation is the generalisation for n for any function of the hierarchical approachany function of the hierarchical approach
Gradient mosaic image
Hierarchical image Waterfall image
The minimal valuation of the catchment The minimal valuation of the catchment basin corresponds to the height of the basin corresponds to the height of the lower FOZ This valuation produces the lower FOZ This valuation produces the same result as the reconstruction of the same result as the reconstruction of the gradient mosaic function by the lower gradient mosaic function by the lower FOZFOZ
WATERFALLS AND MOSAIC IMAGESWATERFALLS AND MOSAIC IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 65
Arcs of minimal heightArcs of minimal heightIn the mosaic image each arc In the mosaic image each arc ccijijseparates two catchment basins separates two catchment basins CBCBiiand and CBCBjj The valuation The valuation vvijij of the arc of the arc is given byis given by
vvijij = | = | ggii -- ggjj ||
where where ggii and and ggjj are the grey values are the grey values in the catchment basinsin the catchment basinsAn arc An arc ccijij is said to be minimal if its is said to be minimal if its valuation is lower than those of all valuation is lower than those of all the other arcs surrounding the other arcs surrounding CBCBiiand and CBCBjj
GRAPH DEFINITIONGRAPH DEFINITION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 66
Definition of a new graphDefinition of a new graph
bullbull its vertices correspond to the arcs of the its vertices correspond to the arcs of the gradient mosaicgradient mosaicbullbull its edges link all arcs surrounding the same its edges link all arcs surrounding the same catchment basincatchment basinbullbull each vertex is valued by the arc valuation as each vertex is valued by the arc valuation as defined in the gradient mosaic defined in the gradient mosaic
In this representation the arcs surrounding In this representation the arcs surrounding the same catchment basin are adjacent the same catchment basin are adjacent Therefore minimal arcs can be connected Therefore minimal arcs can be connected although it is not the case in the gradient although it is not the case in the gradient mosaic as illustrated above (yellow summits mosaic as illustrated above (yellow summits correspond to minimal arcs)correspond to minimal arcs)
GRAPH DEFINITION AND GRAPH DEFINITION AND ASSOCIATED WATERSHEDASSOCIATED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 67
Flooding step 1 (in blue)
The most significant contours of the The most significant contours of the mosaic image correspond to those mosaic image correspond to those separating regions marked by separating regions marked by minimal arcs They are the watershed minimal arcs They are the watershed lines of the watershed transform lines of the watershed transform defined on the previously defined defined on the previously defined graphgraph
Step 2 two CBs in blue amp green Step 3 first dams in red
GRAPH DEFINITION AND ASSOCIATED GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)WATERSHED (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 68
Step 4 Final watershed
Arcs of the gradient mosaic corresponding to the watershed lines
GRAPH DEFINITION AND ASSOCIATED GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)WATERSHED (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 69
The previously defined graph is a 3D The previously defined graph is a 3D valued graph which is not very handyvalued graph which is not very handy
This graph can be transformed into a This graph can be transformed into a planar one by the following procedureplanar one by the following procedurebullbull A new vertex is added in each A new vertex is added in each catchment basincatchment basinbullbull The previous edges are replaced by two The previous edges are replaced by two successive edges linking the original successive edges linking the original vertices through the new onevertices through the new onebull The valuation of the new vertex is given byThe valuation of the new vertex is given by
min (min (vvijij))where where vvijij are the valuations of the arcs surrounding the catchment basinare the valuations of the arcs surrounding the catchment basinij
FROM A 3D to A PLANAR GRAPHFROM A 3D to A PLANAR GRAPH
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 70
The hierarchical imageThe hierarchical image
mosaic
gradient mosaic
hierarchical image
An image named hierarchical image can be An image named hierarchical image can be build from the planar graph The catchment build from the planar graph The catchment basins of the gradient mosaic are filled with basins of the gradient mosaic are filled with grey values corresponding to the valuation of grey values corresponding to the valuation of the new added verticesthe new added verticesThe watersheds of this hierarchical image The watersheds of this hierarchical image give the higher level of hierarchy (with some give the higher level of hierarchy (with some restrictions)restrictions)
IMAGE REPRESENTATIONIMAGE REPRESENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 71
AlsoAlso calledcalled improperlyimproperly saddlesaddle zoneszones(the FOZ notion has (the FOZ notion has nothingnothing to do to do withwith the the classicalclassical saddlesaddle zones Itzones Itrsquorsquos not a s not a local notion and as the WTS local notion and as the WTS therethere isis no no wayway to know a priori if a to know a priori if a givengiven point point of the image of the image belongsbelongs or not to a FOZ)or not to a FOZ)
LowerLower catchmentcatchment basinbasinIt It isis the part of the the part of the catchmentcatchment basin basin whichwhich isis floodedflooded beforebefore the first the first overflowoverflow((whichwhich occursoccurs throughthrough the the lowestlowest FOZ)FOZ)
First overflows
FIRST OVERFLOW ZONES (FOZ)FIRST OVERFLOW ZONES (FOZ)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 72
IntroductionIntroductionConsider the function f and its Consider the function f and its watershed Various catchment watershed Various catchment basins are numbered from 1 to 9 basins are numbered from 1 to 9 Consider the flooding from the Consider the flooding from the minimum m minimum m
When filling CB1 an overflow When filling CB1 an overflow occurs towards CB2occurs towards CB2
Now if we fill in CB2 the first Now if we fill in CB2 the first overflow occurs towards CB1overflow occurs towards CB1
In this case overflows In this case overflows (waterfalls) are symmetrical(waterfalls) are symmetrical
Therefore the part of the Therefore the part of the watershed line separating CB1 watershed line separating CB1 from CB2 can be removed and from CB2 can be removed and the floods in CB1 and CB2 can the floods in CB1 and CB2 can be mergedbe merged
1
WATERFALLSWATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 73
If this flooding process is iterated If this flooding process is iterated the flood invades CB3 which in the flood invades CB3 which in return when flooded pours into the return when flooded pours into the merged basins CB1 and CB2 Here merged basins CB1 and CB2 Here again the waterfalls being again the waterfalls being symmetrical CB3 is merged to the symmetrical CB3 is merged to the floodflood
Step by step because in each case Step by step because in each case waterfalls are symmetrical all the waterfalls are symmetrical all the catchment basins from 1 to 6 are catchment basins from 1 to 6 are mergedmerged
But when the flood pours into CB7 But when the flood pours into CB7 the situation changes Now if we the situation changes Now if we flood CB7 the waterfall is no longer flood CB7 the waterfall is no longer symmetrical Therefore the symmetrical Therefore the watershed line between CB7 and the watershed line between CB7 and the merged basins must be preservedmerged basins must be preserved
WATERFALLS (2)WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 74
Floodingfrom here
The previous process does not The previous process does not work if we start from any work if we start from any basin However the flooding in basin However the flooding in the end reaches the significant the end reaches the significant CBsCBs
The successive floods generate the lower The successive floods generate the lower catchment basins associated with each CB catchment basins associated with each CB (flood just before the overflow through the (flood just before the overflow through the lower saddle zone)lower saddle zone)
This can be achieved directly by a dual This can be achieved directly by a dual reconstruction of the initial function by the reconstruction of the initial function by the lower saddle zones lower saddle zones
SIGNIFICANT CBARCS AND RECONSTRUCTIONSIGNIFICANT CBARCS AND RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 75
InsteadInstead of of usingusing FOZ (not FOZ (not easyeasy to to detectdetect themthem) the ) the wholewhole set of set of watershedwatershed lineslines maymay bebe usedused The The resultresult willwill bebe the the samesame because the because the FOZ FOZ isis the the regionregion atat the the lowestlowest altitude altitude borderingbordering the the catchmentcatchmentbasinbasin
bullbull f initial f initial functionfunctionbullbull let us let us definedefine ggg (x) = f (x) if and g (x) = f (x) if and onlyonly if x if x belongsbelongs to the to the watershedswatersheds of fof fg (x) = max if notg (x) = max if notbullbull h = Rh = Rf f (g) (g) resultresult of the dual of the dual reconstruction of f by g reconstruction of f by g isis alsoalso calledcalledhierarchicalhierarchical imageimagebullbull W(h) W(h) watershedwatershed of h of h producesproduces the the hierarchicalhierarchical segmentation of segmentation of higherhigher levellevel
WhenWhen f f isis a a valuedvalued WTS WTS thisthis hierarchicalhierarchical image image isis identicalidentical to to the the previousprevious definitiondefinition
Lower catchmentbasins
RECONSTRUCTION AND HIERARCHICAL IMAGERECONSTRUCTION AND HIERARCHICAL IMAGE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 76
In this case the hierarchical approach and the waterfall approaIn this case the hierarchical approach and the waterfall approach ch are identical The waterfall transformation is the generalisatioare identical The waterfall transformation is the generalisation for n for any function of the hierarchical approachany function of the hierarchical approach
Gradient mosaic image
Hierarchical image Waterfall image
The minimal valuation of the catchment The minimal valuation of the catchment basin corresponds to the height of the basin corresponds to the height of the lower FOZ This valuation produces the lower FOZ This valuation produces the same result as the reconstruction of the same result as the reconstruction of the gradient mosaic function by the lower gradient mosaic function by the lower FOZFOZ
WATERFALLS AND MOSAIC IMAGESWATERFALLS AND MOSAIC IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 66
Definition of a new graphDefinition of a new graph
bullbull its vertices correspond to the arcs of the its vertices correspond to the arcs of the gradient mosaicgradient mosaicbullbull its edges link all arcs surrounding the same its edges link all arcs surrounding the same catchment basincatchment basinbullbull each vertex is valued by the arc valuation as each vertex is valued by the arc valuation as defined in the gradient mosaic defined in the gradient mosaic
In this representation the arcs surrounding In this representation the arcs surrounding the same catchment basin are adjacent the same catchment basin are adjacent Therefore minimal arcs can be connected Therefore minimal arcs can be connected although it is not the case in the gradient although it is not the case in the gradient mosaic as illustrated above (yellow summits mosaic as illustrated above (yellow summits correspond to minimal arcs)correspond to minimal arcs)
GRAPH DEFINITION AND GRAPH DEFINITION AND ASSOCIATED WATERSHEDASSOCIATED WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 67
Flooding step 1 (in blue)
The most significant contours of the The most significant contours of the mosaic image correspond to those mosaic image correspond to those separating regions marked by separating regions marked by minimal arcs They are the watershed minimal arcs They are the watershed lines of the watershed transform lines of the watershed transform defined on the previously defined defined on the previously defined graphgraph
Step 2 two CBs in blue amp green Step 3 first dams in red
GRAPH DEFINITION AND ASSOCIATED GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)WATERSHED (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 68
Step 4 Final watershed
Arcs of the gradient mosaic corresponding to the watershed lines
GRAPH DEFINITION AND ASSOCIATED GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)WATERSHED (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 69
The previously defined graph is a 3D The previously defined graph is a 3D valued graph which is not very handyvalued graph which is not very handy
This graph can be transformed into a This graph can be transformed into a planar one by the following procedureplanar one by the following procedurebullbull A new vertex is added in each A new vertex is added in each catchment basincatchment basinbullbull The previous edges are replaced by two The previous edges are replaced by two successive edges linking the original successive edges linking the original vertices through the new onevertices through the new onebull The valuation of the new vertex is given byThe valuation of the new vertex is given by
min (min (vvijij))where where vvijij are the valuations of the arcs surrounding the catchment basinare the valuations of the arcs surrounding the catchment basinij
FROM A 3D to A PLANAR GRAPHFROM A 3D to A PLANAR GRAPH
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 70
The hierarchical imageThe hierarchical image
mosaic
gradient mosaic
hierarchical image
An image named hierarchical image can be An image named hierarchical image can be build from the planar graph The catchment build from the planar graph The catchment basins of the gradient mosaic are filled with basins of the gradient mosaic are filled with grey values corresponding to the valuation of grey values corresponding to the valuation of the new added verticesthe new added verticesThe watersheds of this hierarchical image The watersheds of this hierarchical image give the higher level of hierarchy (with some give the higher level of hierarchy (with some restrictions)restrictions)
IMAGE REPRESENTATIONIMAGE REPRESENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 71
AlsoAlso calledcalled improperlyimproperly saddlesaddle zoneszones(the FOZ notion has (the FOZ notion has nothingnothing to do to do withwith the the classicalclassical saddlesaddle zones Itzones Itrsquorsquos not a s not a local notion and as the WTS local notion and as the WTS therethere isis no no wayway to know a priori if a to know a priori if a givengiven point point of the image of the image belongsbelongs or not to a FOZ)or not to a FOZ)
LowerLower catchmentcatchment basinbasinIt It isis the part of the the part of the catchmentcatchment basin basin whichwhich isis floodedflooded beforebefore the first the first overflowoverflow((whichwhich occursoccurs throughthrough the the lowestlowest FOZ)FOZ)
First overflows
FIRST OVERFLOW ZONES (FOZ)FIRST OVERFLOW ZONES (FOZ)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 72
IntroductionIntroductionConsider the function f and its Consider the function f and its watershed Various catchment watershed Various catchment basins are numbered from 1 to 9 basins are numbered from 1 to 9 Consider the flooding from the Consider the flooding from the minimum m minimum m
When filling CB1 an overflow When filling CB1 an overflow occurs towards CB2occurs towards CB2
Now if we fill in CB2 the first Now if we fill in CB2 the first overflow occurs towards CB1overflow occurs towards CB1
In this case overflows In this case overflows (waterfalls) are symmetrical(waterfalls) are symmetrical
Therefore the part of the Therefore the part of the watershed line separating CB1 watershed line separating CB1 from CB2 can be removed and from CB2 can be removed and the floods in CB1 and CB2 can the floods in CB1 and CB2 can be mergedbe merged
1
WATERFALLSWATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 73
If this flooding process is iterated If this flooding process is iterated the flood invades CB3 which in the flood invades CB3 which in return when flooded pours into the return when flooded pours into the merged basins CB1 and CB2 Here merged basins CB1 and CB2 Here again the waterfalls being again the waterfalls being symmetrical CB3 is merged to the symmetrical CB3 is merged to the floodflood
Step by step because in each case Step by step because in each case waterfalls are symmetrical all the waterfalls are symmetrical all the catchment basins from 1 to 6 are catchment basins from 1 to 6 are mergedmerged
But when the flood pours into CB7 But when the flood pours into CB7 the situation changes Now if we the situation changes Now if we flood CB7 the waterfall is no longer flood CB7 the waterfall is no longer symmetrical Therefore the symmetrical Therefore the watershed line between CB7 and the watershed line between CB7 and the merged basins must be preservedmerged basins must be preserved
WATERFALLS (2)WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 74
Floodingfrom here
The previous process does not The previous process does not work if we start from any work if we start from any basin However the flooding in basin However the flooding in the end reaches the significant the end reaches the significant CBsCBs
The successive floods generate the lower The successive floods generate the lower catchment basins associated with each CB catchment basins associated with each CB (flood just before the overflow through the (flood just before the overflow through the lower saddle zone)lower saddle zone)
This can be achieved directly by a dual This can be achieved directly by a dual reconstruction of the initial function by the reconstruction of the initial function by the lower saddle zones lower saddle zones
SIGNIFICANT CBARCS AND RECONSTRUCTIONSIGNIFICANT CBARCS AND RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 75
InsteadInstead of of usingusing FOZ (not FOZ (not easyeasy to to detectdetect themthem) the ) the wholewhole set of set of watershedwatershed lineslines maymay bebe usedused The The resultresult willwill bebe the the samesame because the because the FOZ FOZ isis the the regionregion atat the the lowestlowest altitude altitude borderingbordering the the catchmentcatchmentbasinbasin
bullbull f initial f initial functionfunctionbullbull let us let us definedefine ggg (x) = f (x) if and g (x) = f (x) if and onlyonly if x if x belongsbelongs to the to the watershedswatersheds of fof fg (x) = max if notg (x) = max if notbullbull h = Rh = Rf f (g) (g) resultresult of the dual of the dual reconstruction of f by g reconstruction of f by g isis alsoalso calledcalledhierarchicalhierarchical imageimagebullbull W(h) W(h) watershedwatershed of h of h producesproduces the the hierarchicalhierarchical segmentation of segmentation of higherhigher levellevel
WhenWhen f f isis a a valuedvalued WTS WTS thisthis hierarchicalhierarchical image image isis identicalidentical to to the the previousprevious definitiondefinition
Lower catchmentbasins
RECONSTRUCTION AND HIERARCHICAL IMAGERECONSTRUCTION AND HIERARCHICAL IMAGE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 76
In this case the hierarchical approach and the waterfall approaIn this case the hierarchical approach and the waterfall approach ch are identical The waterfall transformation is the generalisatioare identical The waterfall transformation is the generalisation for n for any function of the hierarchical approachany function of the hierarchical approach
Gradient mosaic image
Hierarchical image Waterfall image
The minimal valuation of the catchment The minimal valuation of the catchment basin corresponds to the height of the basin corresponds to the height of the lower FOZ This valuation produces the lower FOZ This valuation produces the same result as the reconstruction of the same result as the reconstruction of the gradient mosaic function by the lower gradient mosaic function by the lower FOZFOZ
WATERFALLS AND MOSAIC IMAGESWATERFALLS AND MOSAIC IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 67
Flooding step 1 (in blue)
The most significant contours of the The most significant contours of the mosaic image correspond to those mosaic image correspond to those separating regions marked by separating regions marked by minimal arcs They are the watershed minimal arcs They are the watershed lines of the watershed transform lines of the watershed transform defined on the previously defined defined on the previously defined graphgraph
Step 2 two CBs in blue amp green Step 3 first dams in red
GRAPH DEFINITION AND ASSOCIATED GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)WATERSHED (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 68
Step 4 Final watershed
Arcs of the gradient mosaic corresponding to the watershed lines
GRAPH DEFINITION AND ASSOCIATED GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)WATERSHED (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 69
The previously defined graph is a 3D The previously defined graph is a 3D valued graph which is not very handyvalued graph which is not very handy
This graph can be transformed into a This graph can be transformed into a planar one by the following procedureplanar one by the following procedurebullbull A new vertex is added in each A new vertex is added in each catchment basincatchment basinbullbull The previous edges are replaced by two The previous edges are replaced by two successive edges linking the original successive edges linking the original vertices through the new onevertices through the new onebull The valuation of the new vertex is given byThe valuation of the new vertex is given by
min (min (vvijij))where where vvijij are the valuations of the arcs surrounding the catchment basinare the valuations of the arcs surrounding the catchment basinij
FROM A 3D to A PLANAR GRAPHFROM A 3D to A PLANAR GRAPH
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 70
The hierarchical imageThe hierarchical image
mosaic
gradient mosaic
hierarchical image
An image named hierarchical image can be An image named hierarchical image can be build from the planar graph The catchment build from the planar graph The catchment basins of the gradient mosaic are filled with basins of the gradient mosaic are filled with grey values corresponding to the valuation of grey values corresponding to the valuation of the new added verticesthe new added verticesThe watersheds of this hierarchical image The watersheds of this hierarchical image give the higher level of hierarchy (with some give the higher level of hierarchy (with some restrictions)restrictions)
IMAGE REPRESENTATIONIMAGE REPRESENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 71
AlsoAlso calledcalled improperlyimproperly saddlesaddle zoneszones(the FOZ notion has (the FOZ notion has nothingnothing to do to do withwith the the classicalclassical saddlesaddle zones Itzones Itrsquorsquos not a s not a local notion and as the WTS local notion and as the WTS therethere isis no no wayway to know a priori if a to know a priori if a givengiven point point of the image of the image belongsbelongs or not to a FOZ)or not to a FOZ)
LowerLower catchmentcatchment basinbasinIt It isis the part of the the part of the catchmentcatchment basin basin whichwhich isis floodedflooded beforebefore the first the first overflowoverflow((whichwhich occursoccurs throughthrough the the lowestlowest FOZ)FOZ)
First overflows
FIRST OVERFLOW ZONES (FOZ)FIRST OVERFLOW ZONES (FOZ)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 72
IntroductionIntroductionConsider the function f and its Consider the function f and its watershed Various catchment watershed Various catchment basins are numbered from 1 to 9 basins are numbered from 1 to 9 Consider the flooding from the Consider the flooding from the minimum m minimum m
When filling CB1 an overflow When filling CB1 an overflow occurs towards CB2occurs towards CB2
Now if we fill in CB2 the first Now if we fill in CB2 the first overflow occurs towards CB1overflow occurs towards CB1
In this case overflows In this case overflows (waterfalls) are symmetrical(waterfalls) are symmetrical
Therefore the part of the Therefore the part of the watershed line separating CB1 watershed line separating CB1 from CB2 can be removed and from CB2 can be removed and the floods in CB1 and CB2 can the floods in CB1 and CB2 can be mergedbe merged
1
WATERFALLSWATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 73
If this flooding process is iterated If this flooding process is iterated the flood invades CB3 which in the flood invades CB3 which in return when flooded pours into the return when flooded pours into the merged basins CB1 and CB2 Here merged basins CB1 and CB2 Here again the waterfalls being again the waterfalls being symmetrical CB3 is merged to the symmetrical CB3 is merged to the floodflood
Step by step because in each case Step by step because in each case waterfalls are symmetrical all the waterfalls are symmetrical all the catchment basins from 1 to 6 are catchment basins from 1 to 6 are mergedmerged
But when the flood pours into CB7 But when the flood pours into CB7 the situation changes Now if we the situation changes Now if we flood CB7 the waterfall is no longer flood CB7 the waterfall is no longer symmetrical Therefore the symmetrical Therefore the watershed line between CB7 and the watershed line between CB7 and the merged basins must be preservedmerged basins must be preserved
WATERFALLS (2)WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 74
Floodingfrom here
The previous process does not The previous process does not work if we start from any work if we start from any basin However the flooding in basin However the flooding in the end reaches the significant the end reaches the significant CBsCBs
The successive floods generate the lower The successive floods generate the lower catchment basins associated with each CB catchment basins associated with each CB (flood just before the overflow through the (flood just before the overflow through the lower saddle zone)lower saddle zone)
This can be achieved directly by a dual This can be achieved directly by a dual reconstruction of the initial function by the reconstruction of the initial function by the lower saddle zones lower saddle zones
SIGNIFICANT CBARCS AND RECONSTRUCTIONSIGNIFICANT CBARCS AND RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 75
InsteadInstead of of usingusing FOZ (not FOZ (not easyeasy to to detectdetect themthem) the ) the wholewhole set of set of watershedwatershed lineslines maymay bebe usedused The The resultresult willwill bebe the the samesame because the because the FOZ FOZ isis the the regionregion atat the the lowestlowest altitude altitude borderingbordering the the catchmentcatchmentbasinbasin
bullbull f initial f initial functionfunctionbullbull let us let us definedefine ggg (x) = f (x) if and g (x) = f (x) if and onlyonly if x if x belongsbelongs to the to the watershedswatersheds of fof fg (x) = max if notg (x) = max if notbullbull h = Rh = Rf f (g) (g) resultresult of the dual of the dual reconstruction of f by g reconstruction of f by g isis alsoalso calledcalledhierarchicalhierarchical imageimagebullbull W(h) W(h) watershedwatershed of h of h producesproduces the the hierarchicalhierarchical segmentation of segmentation of higherhigher levellevel
WhenWhen f f isis a a valuedvalued WTS WTS thisthis hierarchicalhierarchical image image isis identicalidentical to to the the previousprevious definitiondefinition
Lower catchmentbasins
RECONSTRUCTION AND HIERARCHICAL IMAGERECONSTRUCTION AND HIERARCHICAL IMAGE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 76
In this case the hierarchical approach and the waterfall approaIn this case the hierarchical approach and the waterfall approach ch are identical The waterfall transformation is the generalisatioare identical The waterfall transformation is the generalisation for n for any function of the hierarchical approachany function of the hierarchical approach
Gradient mosaic image
Hierarchical image Waterfall image
The minimal valuation of the catchment The minimal valuation of the catchment basin corresponds to the height of the basin corresponds to the height of the lower FOZ This valuation produces the lower FOZ This valuation produces the same result as the reconstruction of the same result as the reconstruction of the gradient mosaic function by the lower gradient mosaic function by the lower FOZFOZ
WATERFALLS AND MOSAIC IMAGESWATERFALLS AND MOSAIC IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 68
Step 4 Final watershed
Arcs of the gradient mosaic corresponding to the watershed lines
GRAPH DEFINITION AND ASSOCIATED GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)WATERSHED (3)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 69
The previously defined graph is a 3D The previously defined graph is a 3D valued graph which is not very handyvalued graph which is not very handy
This graph can be transformed into a This graph can be transformed into a planar one by the following procedureplanar one by the following procedurebullbull A new vertex is added in each A new vertex is added in each catchment basincatchment basinbullbull The previous edges are replaced by two The previous edges are replaced by two successive edges linking the original successive edges linking the original vertices through the new onevertices through the new onebull The valuation of the new vertex is given byThe valuation of the new vertex is given by
min (min (vvijij))where where vvijij are the valuations of the arcs surrounding the catchment basinare the valuations of the arcs surrounding the catchment basinij
FROM A 3D to A PLANAR GRAPHFROM A 3D to A PLANAR GRAPH
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 70
The hierarchical imageThe hierarchical image
mosaic
gradient mosaic
hierarchical image
An image named hierarchical image can be An image named hierarchical image can be build from the planar graph The catchment build from the planar graph The catchment basins of the gradient mosaic are filled with basins of the gradient mosaic are filled with grey values corresponding to the valuation of grey values corresponding to the valuation of the new added verticesthe new added verticesThe watersheds of this hierarchical image The watersheds of this hierarchical image give the higher level of hierarchy (with some give the higher level of hierarchy (with some restrictions)restrictions)
IMAGE REPRESENTATIONIMAGE REPRESENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 71
AlsoAlso calledcalled improperlyimproperly saddlesaddle zoneszones(the FOZ notion has (the FOZ notion has nothingnothing to do to do withwith the the classicalclassical saddlesaddle zones Itzones Itrsquorsquos not a s not a local notion and as the WTS local notion and as the WTS therethere isis no no wayway to know a priori if a to know a priori if a givengiven point point of the image of the image belongsbelongs or not to a FOZ)or not to a FOZ)
LowerLower catchmentcatchment basinbasinIt It isis the part of the the part of the catchmentcatchment basin basin whichwhich isis floodedflooded beforebefore the first the first overflowoverflow((whichwhich occursoccurs throughthrough the the lowestlowest FOZ)FOZ)
First overflows
FIRST OVERFLOW ZONES (FOZ)FIRST OVERFLOW ZONES (FOZ)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 72
IntroductionIntroductionConsider the function f and its Consider the function f and its watershed Various catchment watershed Various catchment basins are numbered from 1 to 9 basins are numbered from 1 to 9 Consider the flooding from the Consider the flooding from the minimum m minimum m
When filling CB1 an overflow When filling CB1 an overflow occurs towards CB2occurs towards CB2
Now if we fill in CB2 the first Now if we fill in CB2 the first overflow occurs towards CB1overflow occurs towards CB1
In this case overflows In this case overflows (waterfalls) are symmetrical(waterfalls) are symmetrical
Therefore the part of the Therefore the part of the watershed line separating CB1 watershed line separating CB1 from CB2 can be removed and from CB2 can be removed and the floods in CB1 and CB2 can the floods in CB1 and CB2 can be mergedbe merged
1
WATERFALLSWATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 73
If this flooding process is iterated If this flooding process is iterated the flood invades CB3 which in the flood invades CB3 which in return when flooded pours into the return when flooded pours into the merged basins CB1 and CB2 Here merged basins CB1 and CB2 Here again the waterfalls being again the waterfalls being symmetrical CB3 is merged to the symmetrical CB3 is merged to the floodflood
Step by step because in each case Step by step because in each case waterfalls are symmetrical all the waterfalls are symmetrical all the catchment basins from 1 to 6 are catchment basins from 1 to 6 are mergedmerged
But when the flood pours into CB7 But when the flood pours into CB7 the situation changes Now if we the situation changes Now if we flood CB7 the waterfall is no longer flood CB7 the waterfall is no longer symmetrical Therefore the symmetrical Therefore the watershed line between CB7 and the watershed line between CB7 and the merged basins must be preservedmerged basins must be preserved
WATERFALLS (2)WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 74
Floodingfrom here
The previous process does not The previous process does not work if we start from any work if we start from any basin However the flooding in basin However the flooding in the end reaches the significant the end reaches the significant CBsCBs
The successive floods generate the lower The successive floods generate the lower catchment basins associated with each CB catchment basins associated with each CB (flood just before the overflow through the (flood just before the overflow through the lower saddle zone)lower saddle zone)
This can be achieved directly by a dual This can be achieved directly by a dual reconstruction of the initial function by the reconstruction of the initial function by the lower saddle zones lower saddle zones
SIGNIFICANT CBARCS AND RECONSTRUCTIONSIGNIFICANT CBARCS AND RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 75
InsteadInstead of of usingusing FOZ (not FOZ (not easyeasy to to detectdetect themthem) the ) the wholewhole set of set of watershedwatershed lineslines maymay bebe usedused The The resultresult willwill bebe the the samesame because the because the FOZ FOZ isis the the regionregion atat the the lowestlowest altitude altitude borderingbordering the the catchmentcatchmentbasinbasin
bullbull f initial f initial functionfunctionbullbull let us let us definedefine ggg (x) = f (x) if and g (x) = f (x) if and onlyonly if x if x belongsbelongs to the to the watershedswatersheds of fof fg (x) = max if notg (x) = max if notbullbull h = Rh = Rf f (g) (g) resultresult of the dual of the dual reconstruction of f by g reconstruction of f by g isis alsoalso calledcalledhierarchicalhierarchical imageimagebullbull W(h) W(h) watershedwatershed of h of h producesproduces the the hierarchicalhierarchical segmentation of segmentation of higherhigher levellevel
WhenWhen f f isis a a valuedvalued WTS WTS thisthis hierarchicalhierarchical image image isis identicalidentical to to the the previousprevious definitiondefinition
Lower catchmentbasins
RECONSTRUCTION AND HIERARCHICAL IMAGERECONSTRUCTION AND HIERARCHICAL IMAGE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 76
In this case the hierarchical approach and the waterfall approaIn this case the hierarchical approach and the waterfall approach ch are identical The waterfall transformation is the generalisatioare identical The waterfall transformation is the generalisation for n for any function of the hierarchical approachany function of the hierarchical approach
Gradient mosaic image
Hierarchical image Waterfall image
The minimal valuation of the catchment The minimal valuation of the catchment basin corresponds to the height of the basin corresponds to the height of the lower FOZ This valuation produces the lower FOZ This valuation produces the same result as the reconstruction of the same result as the reconstruction of the gradient mosaic function by the lower gradient mosaic function by the lower FOZFOZ
WATERFALLS AND MOSAIC IMAGESWATERFALLS AND MOSAIC IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 69
The previously defined graph is a 3D The previously defined graph is a 3D valued graph which is not very handyvalued graph which is not very handy
This graph can be transformed into a This graph can be transformed into a planar one by the following procedureplanar one by the following procedurebullbull A new vertex is added in each A new vertex is added in each catchment basincatchment basinbullbull The previous edges are replaced by two The previous edges are replaced by two successive edges linking the original successive edges linking the original vertices through the new onevertices through the new onebull The valuation of the new vertex is given byThe valuation of the new vertex is given by
min (min (vvijij))where where vvijij are the valuations of the arcs surrounding the catchment basinare the valuations of the arcs surrounding the catchment basinij
FROM A 3D to A PLANAR GRAPHFROM A 3D to A PLANAR GRAPH
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 70
The hierarchical imageThe hierarchical image
mosaic
gradient mosaic
hierarchical image
An image named hierarchical image can be An image named hierarchical image can be build from the planar graph The catchment build from the planar graph The catchment basins of the gradient mosaic are filled with basins of the gradient mosaic are filled with grey values corresponding to the valuation of grey values corresponding to the valuation of the new added verticesthe new added verticesThe watersheds of this hierarchical image The watersheds of this hierarchical image give the higher level of hierarchy (with some give the higher level of hierarchy (with some restrictions)restrictions)
IMAGE REPRESENTATIONIMAGE REPRESENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 71
AlsoAlso calledcalled improperlyimproperly saddlesaddle zoneszones(the FOZ notion has (the FOZ notion has nothingnothing to do to do withwith the the classicalclassical saddlesaddle zones Itzones Itrsquorsquos not a s not a local notion and as the WTS local notion and as the WTS therethere isis no no wayway to know a priori if a to know a priori if a givengiven point point of the image of the image belongsbelongs or not to a FOZ)or not to a FOZ)
LowerLower catchmentcatchment basinbasinIt It isis the part of the the part of the catchmentcatchment basin basin whichwhich isis floodedflooded beforebefore the first the first overflowoverflow((whichwhich occursoccurs throughthrough the the lowestlowest FOZ)FOZ)
First overflows
FIRST OVERFLOW ZONES (FOZ)FIRST OVERFLOW ZONES (FOZ)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 72
IntroductionIntroductionConsider the function f and its Consider the function f and its watershed Various catchment watershed Various catchment basins are numbered from 1 to 9 basins are numbered from 1 to 9 Consider the flooding from the Consider the flooding from the minimum m minimum m
When filling CB1 an overflow When filling CB1 an overflow occurs towards CB2occurs towards CB2
Now if we fill in CB2 the first Now if we fill in CB2 the first overflow occurs towards CB1overflow occurs towards CB1
In this case overflows In this case overflows (waterfalls) are symmetrical(waterfalls) are symmetrical
Therefore the part of the Therefore the part of the watershed line separating CB1 watershed line separating CB1 from CB2 can be removed and from CB2 can be removed and the floods in CB1 and CB2 can the floods in CB1 and CB2 can be mergedbe merged
1
WATERFALLSWATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 73
If this flooding process is iterated If this flooding process is iterated the flood invades CB3 which in the flood invades CB3 which in return when flooded pours into the return when flooded pours into the merged basins CB1 and CB2 Here merged basins CB1 and CB2 Here again the waterfalls being again the waterfalls being symmetrical CB3 is merged to the symmetrical CB3 is merged to the floodflood
Step by step because in each case Step by step because in each case waterfalls are symmetrical all the waterfalls are symmetrical all the catchment basins from 1 to 6 are catchment basins from 1 to 6 are mergedmerged
But when the flood pours into CB7 But when the flood pours into CB7 the situation changes Now if we the situation changes Now if we flood CB7 the waterfall is no longer flood CB7 the waterfall is no longer symmetrical Therefore the symmetrical Therefore the watershed line between CB7 and the watershed line between CB7 and the merged basins must be preservedmerged basins must be preserved
WATERFALLS (2)WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 74
Floodingfrom here
The previous process does not The previous process does not work if we start from any work if we start from any basin However the flooding in basin However the flooding in the end reaches the significant the end reaches the significant CBsCBs
The successive floods generate the lower The successive floods generate the lower catchment basins associated with each CB catchment basins associated with each CB (flood just before the overflow through the (flood just before the overflow through the lower saddle zone)lower saddle zone)
This can be achieved directly by a dual This can be achieved directly by a dual reconstruction of the initial function by the reconstruction of the initial function by the lower saddle zones lower saddle zones
SIGNIFICANT CBARCS AND RECONSTRUCTIONSIGNIFICANT CBARCS AND RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 75
InsteadInstead of of usingusing FOZ (not FOZ (not easyeasy to to detectdetect themthem) the ) the wholewhole set of set of watershedwatershed lineslines maymay bebe usedused The The resultresult willwill bebe the the samesame because the because the FOZ FOZ isis the the regionregion atat the the lowestlowest altitude altitude borderingbordering the the catchmentcatchmentbasinbasin
bullbull f initial f initial functionfunctionbullbull let us let us definedefine ggg (x) = f (x) if and g (x) = f (x) if and onlyonly if x if x belongsbelongs to the to the watershedswatersheds of fof fg (x) = max if notg (x) = max if notbullbull h = Rh = Rf f (g) (g) resultresult of the dual of the dual reconstruction of f by g reconstruction of f by g isis alsoalso calledcalledhierarchicalhierarchical imageimagebullbull W(h) W(h) watershedwatershed of h of h producesproduces the the hierarchicalhierarchical segmentation of segmentation of higherhigher levellevel
WhenWhen f f isis a a valuedvalued WTS WTS thisthis hierarchicalhierarchical image image isis identicalidentical to to the the previousprevious definitiondefinition
Lower catchmentbasins
RECONSTRUCTION AND HIERARCHICAL IMAGERECONSTRUCTION AND HIERARCHICAL IMAGE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 76
In this case the hierarchical approach and the waterfall approaIn this case the hierarchical approach and the waterfall approach ch are identical The waterfall transformation is the generalisatioare identical The waterfall transformation is the generalisation for n for any function of the hierarchical approachany function of the hierarchical approach
Gradient mosaic image
Hierarchical image Waterfall image
The minimal valuation of the catchment The minimal valuation of the catchment basin corresponds to the height of the basin corresponds to the height of the lower FOZ This valuation produces the lower FOZ This valuation produces the same result as the reconstruction of the same result as the reconstruction of the gradient mosaic function by the lower gradient mosaic function by the lower FOZFOZ
WATERFALLS AND MOSAIC IMAGESWATERFALLS AND MOSAIC IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 70
The hierarchical imageThe hierarchical image
mosaic
gradient mosaic
hierarchical image
An image named hierarchical image can be An image named hierarchical image can be build from the planar graph The catchment build from the planar graph The catchment basins of the gradient mosaic are filled with basins of the gradient mosaic are filled with grey values corresponding to the valuation of grey values corresponding to the valuation of the new added verticesthe new added verticesThe watersheds of this hierarchical image The watersheds of this hierarchical image give the higher level of hierarchy (with some give the higher level of hierarchy (with some restrictions)restrictions)
IMAGE REPRESENTATIONIMAGE REPRESENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 71
AlsoAlso calledcalled improperlyimproperly saddlesaddle zoneszones(the FOZ notion has (the FOZ notion has nothingnothing to do to do withwith the the classicalclassical saddlesaddle zones Itzones Itrsquorsquos not a s not a local notion and as the WTS local notion and as the WTS therethere isis no no wayway to know a priori if a to know a priori if a givengiven point point of the image of the image belongsbelongs or not to a FOZ)or not to a FOZ)
LowerLower catchmentcatchment basinbasinIt It isis the part of the the part of the catchmentcatchment basin basin whichwhich isis floodedflooded beforebefore the first the first overflowoverflow((whichwhich occursoccurs throughthrough the the lowestlowest FOZ)FOZ)
First overflows
FIRST OVERFLOW ZONES (FOZ)FIRST OVERFLOW ZONES (FOZ)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 72
IntroductionIntroductionConsider the function f and its Consider the function f and its watershed Various catchment watershed Various catchment basins are numbered from 1 to 9 basins are numbered from 1 to 9 Consider the flooding from the Consider the flooding from the minimum m minimum m
When filling CB1 an overflow When filling CB1 an overflow occurs towards CB2occurs towards CB2
Now if we fill in CB2 the first Now if we fill in CB2 the first overflow occurs towards CB1overflow occurs towards CB1
In this case overflows In this case overflows (waterfalls) are symmetrical(waterfalls) are symmetrical
Therefore the part of the Therefore the part of the watershed line separating CB1 watershed line separating CB1 from CB2 can be removed and from CB2 can be removed and the floods in CB1 and CB2 can the floods in CB1 and CB2 can be mergedbe merged
1
WATERFALLSWATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 73
If this flooding process is iterated If this flooding process is iterated the flood invades CB3 which in the flood invades CB3 which in return when flooded pours into the return when flooded pours into the merged basins CB1 and CB2 Here merged basins CB1 and CB2 Here again the waterfalls being again the waterfalls being symmetrical CB3 is merged to the symmetrical CB3 is merged to the floodflood
Step by step because in each case Step by step because in each case waterfalls are symmetrical all the waterfalls are symmetrical all the catchment basins from 1 to 6 are catchment basins from 1 to 6 are mergedmerged
But when the flood pours into CB7 But when the flood pours into CB7 the situation changes Now if we the situation changes Now if we flood CB7 the waterfall is no longer flood CB7 the waterfall is no longer symmetrical Therefore the symmetrical Therefore the watershed line between CB7 and the watershed line between CB7 and the merged basins must be preservedmerged basins must be preserved
WATERFALLS (2)WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 74
Floodingfrom here
The previous process does not The previous process does not work if we start from any work if we start from any basin However the flooding in basin However the flooding in the end reaches the significant the end reaches the significant CBsCBs
The successive floods generate the lower The successive floods generate the lower catchment basins associated with each CB catchment basins associated with each CB (flood just before the overflow through the (flood just before the overflow through the lower saddle zone)lower saddle zone)
This can be achieved directly by a dual This can be achieved directly by a dual reconstruction of the initial function by the reconstruction of the initial function by the lower saddle zones lower saddle zones
SIGNIFICANT CBARCS AND RECONSTRUCTIONSIGNIFICANT CBARCS AND RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 75
InsteadInstead of of usingusing FOZ (not FOZ (not easyeasy to to detectdetect themthem) the ) the wholewhole set of set of watershedwatershed lineslines maymay bebe usedused The The resultresult willwill bebe the the samesame because the because the FOZ FOZ isis the the regionregion atat the the lowestlowest altitude altitude borderingbordering the the catchmentcatchmentbasinbasin
bullbull f initial f initial functionfunctionbullbull let us let us definedefine ggg (x) = f (x) if and g (x) = f (x) if and onlyonly if x if x belongsbelongs to the to the watershedswatersheds of fof fg (x) = max if notg (x) = max if notbullbull h = Rh = Rf f (g) (g) resultresult of the dual of the dual reconstruction of f by g reconstruction of f by g isis alsoalso calledcalledhierarchicalhierarchical imageimagebullbull W(h) W(h) watershedwatershed of h of h producesproduces the the hierarchicalhierarchical segmentation of segmentation of higherhigher levellevel
WhenWhen f f isis a a valuedvalued WTS WTS thisthis hierarchicalhierarchical image image isis identicalidentical to to the the previousprevious definitiondefinition
Lower catchmentbasins
RECONSTRUCTION AND HIERARCHICAL IMAGERECONSTRUCTION AND HIERARCHICAL IMAGE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 76
In this case the hierarchical approach and the waterfall approaIn this case the hierarchical approach and the waterfall approach ch are identical The waterfall transformation is the generalisatioare identical The waterfall transformation is the generalisation for n for any function of the hierarchical approachany function of the hierarchical approach
Gradient mosaic image
Hierarchical image Waterfall image
The minimal valuation of the catchment The minimal valuation of the catchment basin corresponds to the height of the basin corresponds to the height of the lower FOZ This valuation produces the lower FOZ This valuation produces the same result as the reconstruction of the same result as the reconstruction of the gradient mosaic function by the lower gradient mosaic function by the lower FOZFOZ
WATERFALLS AND MOSAIC IMAGESWATERFALLS AND MOSAIC IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 71
AlsoAlso calledcalled improperlyimproperly saddlesaddle zoneszones(the FOZ notion has (the FOZ notion has nothingnothing to do to do withwith the the classicalclassical saddlesaddle zones Itzones Itrsquorsquos not a s not a local notion and as the WTS local notion and as the WTS therethere isis no no wayway to know a priori if a to know a priori if a givengiven point point of the image of the image belongsbelongs or not to a FOZ)or not to a FOZ)
LowerLower catchmentcatchment basinbasinIt It isis the part of the the part of the catchmentcatchment basin basin whichwhich isis floodedflooded beforebefore the first the first overflowoverflow((whichwhich occursoccurs throughthrough the the lowestlowest FOZ)FOZ)
First overflows
FIRST OVERFLOW ZONES (FOZ)FIRST OVERFLOW ZONES (FOZ)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 72
IntroductionIntroductionConsider the function f and its Consider the function f and its watershed Various catchment watershed Various catchment basins are numbered from 1 to 9 basins are numbered from 1 to 9 Consider the flooding from the Consider the flooding from the minimum m minimum m
When filling CB1 an overflow When filling CB1 an overflow occurs towards CB2occurs towards CB2
Now if we fill in CB2 the first Now if we fill in CB2 the first overflow occurs towards CB1overflow occurs towards CB1
In this case overflows In this case overflows (waterfalls) are symmetrical(waterfalls) are symmetrical
Therefore the part of the Therefore the part of the watershed line separating CB1 watershed line separating CB1 from CB2 can be removed and from CB2 can be removed and the floods in CB1 and CB2 can the floods in CB1 and CB2 can be mergedbe merged
1
WATERFALLSWATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 73
If this flooding process is iterated If this flooding process is iterated the flood invades CB3 which in the flood invades CB3 which in return when flooded pours into the return when flooded pours into the merged basins CB1 and CB2 Here merged basins CB1 and CB2 Here again the waterfalls being again the waterfalls being symmetrical CB3 is merged to the symmetrical CB3 is merged to the floodflood
Step by step because in each case Step by step because in each case waterfalls are symmetrical all the waterfalls are symmetrical all the catchment basins from 1 to 6 are catchment basins from 1 to 6 are mergedmerged
But when the flood pours into CB7 But when the flood pours into CB7 the situation changes Now if we the situation changes Now if we flood CB7 the waterfall is no longer flood CB7 the waterfall is no longer symmetrical Therefore the symmetrical Therefore the watershed line between CB7 and the watershed line between CB7 and the merged basins must be preservedmerged basins must be preserved
WATERFALLS (2)WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 74
Floodingfrom here
The previous process does not The previous process does not work if we start from any work if we start from any basin However the flooding in basin However the flooding in the end reaches the significant the end reaches the significant CBsCBs
The successive floods generate the lower The successive floods generate the lower catchment basins associated with each CB catchment basins associated with each CB (flood just before the overflow through the (flood just before the overflow through the lower saddle zone)lower saddle zone)
This can be achieved directly by a dual This can be achieved directly by a dual reconstruction of the initial function by the reconstruction of the initial function by the lower saddle zones lower saddle zones
SIGNIFICANT CBARCS AND RECONSTRUCTIONSIGNIFICANT CBARCS AND RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 75
InsteadInstead of of usingusing FOZ (not FOZ (not easyeasy to to detectdetect themthem) the ) the wholewhole set of set of watershedwatershed lineslines maymay bebe usedused The The resultresult willwill bebe the the samesame because the because the FOZ FOZ isis the the regionregion atat the the lowestlowest altitude altitude borderingbordering the the catchmentcatchmentbasinbasin
bullbull f initial f initial functionfunctionbullbull let us let us definedefine ggg (x) = f (x) if and g (x) = f (x) if and onlyonly if x if x belongsbelongs to the to the watershedswatersheds of fof fg (x) = max if notg (x) = max if notbullbull h = Rh = Rf f (g) (g) resultresult of the dual of the dual reconstruction of f by g reconstruction of f by g isis alsoalso calledcalledhierarchicalhierarchical imageimagebullbull W(h) W(h) watershedwatershed of h of h producesproduces the the hierarchicalhierarchical segmentation of segmentation of higherhigher levellevel
WhenWhen f f isis a a valuedvalued WTS WTS thisthis hierarchicalhierarchical image image isis identicalidentical to to the the previousprevious definitiondefinition
Lower catchmentbasins
RECONSTRUCTION AND HIERARCHICAL IMAGERECONSTRUCTION AND HIERARCHICAL IMAGE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 76
In this case the hierarchical approach and the waterfall approaIn this case the hierarchical approach and the waterfall approach ch are identical The waterfall transformation is the generalisatioare identical The waterfall transformation is the generalisation for n for any function of the hierarchical approachany function of the hierarchical approach
Gradient mosaic image
Hierarchical image Waterfall image
The minimal valuation of the catchment The minimal valuation of the catchment basin corresponds to the height of the basin corresponds to the height of the lower FOZ This valuation produces the lower FOZ This valuation produces the same result as the reconstruction of the same result as the reconstruction of the gradient mosaic function by the lower gradient mosaic function by the lower FOZFOZ
WATERFALLS AND MOSAIC IMAGESWATERFALLS AND MOSAIC IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 72
IntroductionIntroductionConsider the function f and its Consider the function f and its watershed Various catchment watershed Various catchment basins are numbered from 1 to 9 basins are numbered from 1 to 9 Consider the flooding from the Consider the flooding from the minimum m minimum m
When filling CB1 an overflow When filling CB1 an overflow occurs towards CB2occurs towards CB2
Now if we fill in CB2 the first Now if we fill in CB2 the first overflow occurs towards CB1overflow occurs towards CB1
In this case overflows In this case overflows (waterfalls) are symmetrical(waterfalls) are symmetrical
Therefore the part of the Therefore the part of the watershed line separating CB1 watershed line separating CB1 from CB2 can be removed and from CB2 can be removed and the floods in CB1 and CB2 can the floods in CB1 and CB2 can be mergedbe merged
1
WATERFALLSWATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 73
If this flooding process is iterated If this flooding process is iterated the flood invades CB3 which in the flood invades CB3 which in return when flooded pours into the return when flooded pours into the merged basins CB1 and CB2 Here merged basins CB1 and CB2 Here again the waterfalls being again the waterfalls being symmetrical CB3 is merged to the symmetrical CB3 is merged to the floodflood
Step by step because in each case Step by step because in each case waterfalls are symmetrical all the waterfalls are symmetrical all the catchment basins from 1 to 6 are catchment basins from 1 to 6 are mergedmerged
But when the flood pours into CB7 But when the flood pours into CB7 the situation changes Now if we the situation changes Now if we flood CB7 the waterfall is no longer flood CB7 the waterfall is no longer symmetrical Therefore the symmetrical Therefore the watershed line between CB7 and the watershed line between CB7 and the merged basins must be preservedmerged basins must be preserved
WATERFALLS (2)WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 74
Floodingfrom here
The previous process does not The previous process does not work if we start from any work if we start from any basin However the flooding in basin However the flooding in the end reaches the significant the end reaches the significant CBsCBs
The successive floods generate the lower The successive floods generate the lower catchment basins associated with each CB catchment basins associated with each CB (flood just before the overflow through the (flood just before the overflow through the lower saddle zone)lower saddle zone)
This can be achieved directly by a dual This can be achieved directly by a dual reconstruction of the initial function by the reconstruction of the initial function by the lower saddle zones lower saddle zones
SIGNIFICANT CBARCS AND RECONSTRUCTIONSIGNIFICANT CBARCS AND RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 75
InsteadInstead of of usingusing FOZ (not FOZ (not easyeasy to to detectdetect themthem) the ) the wholewhole set of set of watershedwatershed lineslines maymay bebe usedused The The resultresult willwill bebe the the samesame because the because the FOZ FOZ isis the the regionregion atat the the lowestlowest altitude altitude borderingbordering the the catchmentcatchmentbasinbasin
bullbull f initial f initial functionfunctionbullbull let us let us definedefine ggg (x) = f (x) if and g (x) = f (x) if and onlyonly if x if x belongsbelongs to the to the watershedswatersheds of fof fg (x) = max if notg (x) = max if notbullbull h = Rh = Rf f (g) (g) resultresult of the dual of the dual reconstruction of f by g reconstruction of f by g isis alsoalso calledcalledhierarchicalhierarchical imageimagebullbull W(h) W(h) watershedwatershed of h of h producesproduces the the hierarchicalhierarchical segmentation of segmentation of higherhigher levellevel
WhenWhen f f isis a a valuedvalued WTS WTS thisthis hierarchicalhierarchical image image isis identicalidentical to to the the previousprevious definitiondefinition
Lower catchmentbasins
RECONSTRUCTION AND HIERARCHICAL IMAGERECONSTRUCTION AND HIERARCHICAL IMAGE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 76
In this case the hierarchical approach and the waterfall approaIn this case the hierarchical approach and the waterfall approach ch are identical The waterfall transformation is the generalisatioare identical The waterfall transformation is the generalisation for n for any function of the hierarchical approachany function of the hierarchical approach
Gradient mosaic image
Hierarchical image Waterfall image
The minimal valuation of the catchment The minimal valuation of the catchment basin corresponds to the height of the basin corresponds to the height of the lower FOZ This valuation produces the lower FOZ This valuation produces the same result as the reconstruction of the same result as the reconstruction of the gradient mosaic function by the lower gradient mosaic function by the lower FOZFOZ
WATERFALLS AND MOSAIC IMAGESWATERFALLS AND MOSAIC IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 73
If this flooding process is iterated If this flooding process is iterated the flood invades CB3 which in the flood invades CB3 which in return when flooded pours into the return when flooded pours into the merged basins CB1 and CB2 Here merged basins CB1 and CB2 Here again the waterfalls being again the waterfalls being symmetrical CB3 is merged to the symmetrical CB3 is merged to the floodflood
Step by step because in each case Step by step because in each case waterfalls are symmetrical all the waterfalls are symmetrical all the catchment basins from 1 to 6 are catchment basins from 1 to 6 are mergedmerged
But when the flood pours into CB7 But when the flood pours into CB7 the situation changes Now if we the situation changes Now if we flood CB7 the waterfall is no longer flood CB7 the waterfall is no longer symmetrical Therefore the symmetrical Therefore the watershed line between CB7 and the watershed line between CB7 and the merged basins must be preservedmerged basins must be preserved
WATERFALLS (2)WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 74
Floodingfrom here
The previous process does not The previous process does not work if we start from any work if we start from any basin However the flooding in basin However the flooding in the end reaches the significant the end reaches the significant CBsCBs
The successive floods generate the lower The successive floods generate the lower catchment basins associated with each CB catchment basins associated with each CB (flood just before the overflow through the (flood just before the overflow through the lower saddle zone)lower saddle zone)
This can be achieved directly by a dual This can be achieved directly by a dual reconstruction of the initial function by the reconstruction of the initial function by the lower saddle zones lower saddle zones
SIGNIFICANT CBARCS AND RECONSTRUCTIONSIGNIFICANT CBARCS AND RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 75
InsteadInstead of of usingusing FOZ (not FOZ (not easyeasy to to detectdetect themthem) the ) the wholewhole set of set of watershedwatershed lineslines maymay bebe usedused The The resultresult willwill bebe the the samesame because the because the FOZ FOZ isis the the regionregion atat the the lowestlowest altitude altitude borderingbordering the the catchmentcatchmentbasinbasin
bullbull f initial f initial functionfunctionbullbull let us let us definedefine ggg (x) = f (x) if and g (x) = f (x) if and onlyonly if x if x belongsbelongs to the to the watershedswatersheds of fof fg (x) = max if notg (x) = max if notbullbull h = Rh = Rf f (g) (g) resultresult of the dual of the dual reconstruction of f by g reconstruction of f by g isis alsoalso calledcalledhierarchicalhierarchical imageimagebullbull W(h) W(h) watershedwatershed of h of h producesproduces the the hierarchicalhierarchical segmentation of segmentation of higherhigher levellevel
WhenWhen f f isis a a valuedvalued WTS WTS thisthis hierarchicalhierarchical image image isis identicalidentical to to the the previousprevious definitiondefinition
Lower catchmentbasins
RECONSTRUCTION AND HIERARCHICAL IMAGERECONSTRUCTION AND HIERARCHICAL IMAGE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 76
In this case the hierarchical approach and the waterfall approaIn this case the hierarchical approach and the waterfall approach ch are identical The waterfall transformation is the generalisatioare identical The waterfall transformation is the generalisation for n for any function of the hierarchical approachany function of the hierarchical approach
Gradient mosaic image
Hierarchical image Waterfall image
The minimal valuation of the catchment The minimal valuation of the catchment basin corresponds to the height of the basin corresponds to the height of the lower FOZ This valuation produces the lower FOZ This valuation produces the same result as the reconstruction of the same result as the reconstruction of the gradient mosaic function by the lower gradient mosaic function by the lower FOZFOZ
WATERFALLS AND MOSAIC IMAGESWATERFALLS AND MOSAIC IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 74
Floodingfrom here
The previous process does not The previous process does not work if we start from any work if we start from any basin However the flooding in basin However the flooding in the end reaches the significant the end reaches the significant CBsCBs
The successive floods generate the lower The successive floods generate the lower catchment basins associated with each CB catchment basins associated with each CB (flood just before the overflow through the (flood just before the overflow through the lower saddle zone)lower saddle zone)
This can be achieved directly by a dual This can be achieved directly by a dual reconstruction of the initial function by the reconstruction of the initial function by the lower saddle zones lower saddle zones
SIGNIFICANT CBARCS AND RECONSTRUCTIONSIGNIFICANT CBARCS AND RECONSTRUCTION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 75
InsteadInstead of of usingusing FOZ (not FOZ (not easyeasy to to detectdetect themthem) the ) the wholewhole set of set of watershedwatershed lineslines maymay bebe usedused The The resultresult willwill bebe the the samesame because the because the FOZ FOZ isis the the regionregion atat the the lowestlowest altitude altitude borderingbordering the the catchmentcatchmentbasinbasin
bullbull f initial f initial functionfunctionbullbull let us let us definedefine ggg (x) = f (x) if and g (x) = f (x) if and onlyonly if x if x belongsbelongs to the to the watershedswatersheds of fof fg (x) = max if notg (x) = max if notbullbull h = Rh = Rf f (g) (g) resultresult of the dual of the dual reconstruction of f by g reconstruction of f by g isis alsoalso calledcalledhierarchicalhierarchical imageimagebullbull W(h) W(h) watershedwatershed of h of h producesproduces the the hierarchicalhierarchical segmentation of segmentation of higherhigher levellevel
WhenWhen f f isis a a valuedvalued WTS WTS thisthis hierarchicalhierarchical image image isis identicalidentical to to the the previousprevious definitiondefinition
Lower catchmentbasins
RECONSTRUCTION AND HIERARCHICAL IMAGERECONSTRUCTION AND HIERARCHICAL IMAGE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 76
In this case the hierarchical approach and the waterfall approaIn this case the hierarchical approach and the waterfall approach ch are identical The waterfall transformation is the generalisatioare identical The waterfall transformation is the generalisation for n for any function of the hierarchical approachany function of the hierarchical approach
Gradient mosaic image
Hierarchical image Waterfall image
The minimal valuation of the catchment The minimal valuation of the catchment basin corresponds to the height of the basin corresponds to the height of the lower FOZ This valuation produces the lower FOZ This valuation produces the same result as the reconstruction of the same result as the reconstruction of the gradient mosaic function by the lower gradient mosaic function by the lower FOZFOZ
WATERFALLS AND MOSAIC IMAGESWATERFALLS AND MOSAIC IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 75
InsteadInstead of of usingusing FOZ (not FOZ (not easyeasy to to detectdetect themthem) the ) the wholewhole set of set of watershedwatershed lineslines maymay bebe usedused The The resultresult willwill bebe the the samesame because the because the FOZ FOZ isis the the regionregion atat the the lowestlowest altitude altitude borderingbordering the the catchmentcatchmentbasinbasin
bullbull f initial f initial functionfunctionbullbull let us let us definedefine ggg (x) = f (x) if and g (x) = f (x) if and onlyonly if x if x belongsbelongs to the to the watershedswatersheds of fof fg (x) = max if notg (x) = max if notbullbull h = Rh = Rf f (g) (g) resultresult of the dual of the dual reconstruction of f by g reconstruction of f by g isis alsoalso calledcalledhierarchicalhierarchical imageimagebullbull W(h) W(h) watershedwatershed of h of h producesproduces the the hierarchicalhierarchical segmentation of segmentation of higherhigher levellevel
WhenWhen f f isis a a valuedvalued WTS WTS thisthis hierarchicalhierarchical image image isis identicalidentical to to the the previousprevious definitiondefinition
Lower catchmentbasins
RECONSTRUCTION AND HIERARCHICAL IMAGERECONSTRUCTION AND HIERARCHICAL IMAGE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 76
In this case the hierarchical approach and the waterfall approaIn this case the hierarchical approach and the waterfall approach ch are identical The waterfall transformation is the generalisatioare identical The waterfall transformation is the generalisation for n for any function of the hierarchical approachany function of the hierarchical approach
Gradient mosaic image
Hierarchical image Waterfall image
The minimal valuation of the catchment The minimal valuation of the catchment basin corresponds to the height of the basin corresponds to the height of the lower FOZ This valuation produces the lower FOZ This valuation produces the same result as the reconstruction of the same result as the reconstruction of the gradient mosaic function by the lower gradient mosaic function by the lower FOZFOZ
WATERFALLS AND MOSAIC IMAGESWATERFALLS AND MOSAIC IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 76
In this case the hierarchical approach and the waterfall approaIn this case the hierarchical approach and the waterfall approach ch are identical The waterfall transformation is the generalisatioare identical The waterfall transformation is the generalisation for n for any function of the hierarchical approachany function of the hierarchical approach
Gradient mosaic image
Hierarchical image Waterfall image
The minimal valuation of the catchment The minimal valuation of the catchment basin corresponds to the height of the basin corresponds to the height of the lower FOZ This valuation produces the lower FOZ This valuation produces the same result as the reconstruction of the same result as the reconstruction of the gradient mosaic function by the lower gradient mosaic function by the lower FOZFOZ
WATERFALLS AND MOSAIC IMAGESWATERFALLS AND MOSAIC IMAGES
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 77
Original imageOriginal image
Initial watershedInitial watershed
Mosaic imageMosaic image
First level of First level of hierarchyhierarchy
HIERARCHICAL SEGMENTATION EXAMPLEHIERARCHICAL SEGMENTATION EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 78
Protocolbull One One startsstarts fromfrom an initial an initial valuedvalued wateshedwateshed ss00bullbull An An iterativeiterative processprocess producesproduces successive successive hierarchicalhierarchicalsegmentations ssegmentations siissii = w(h= w(hii--11) ) wherewhere hhii--11 isis the the hierarchicalhierarchical image image associatedassociated to the to the segmentation ssegmentation sii--11
HierarchicalHierarchicalsegmentationsegmentation
HierarchicalHierarchical imageimage
Watershed Reconstruction
USE OF WATERFALLSUSE OF WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 79
bull N N hierarchcalhierarchcal levelslevels withwith SSNN = = emptyemptybullbull It It isis difficultdifficult to select a to select a laquolaquo goodgood raquoraquo hierarchicalhierarchical levellevelbullbull OtherOther crucial crucial problemsproblems appearappearhelliphellip
hellip
S0 S1 S2
Initial image f and Initial image f and itsits gradient ggradient g
SS00 = W(g)= W(g)
USE OF WATERFALLS (2)USE OF WATERFALLS (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 80
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 81
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 82
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 83
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 84
bullbull ItItrsquorsquos a non parametric approachs a non parametric approachbullbullThe waterfall can be iterated leading to possible higher levelsThe waterfall can be iterated leading to possible higher levels of of hierarchy buthierarchy buthelliphellipbullbull Stop criterion not availableStop criterion not availablebullbull The successive hierarchical levels are far from being relevantThe successive hierarchical levels are far from being relevantlaquolaquo MyopiaMyopia raquoraquo for for hierarchieshierarchies of of differentdifferent levelslevels (classification (classification errorserrors))
PROBLEMS WITH THE WATERFALLSPROBLEMS WITH THE WATERFALLS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 85
1
HierarchicalHierarchical image himage hi+1 i+1 (in (in redred))
ThreeThree differentdifferent kindskinds of of removedremoved contours contours appearappear11 Contours Contours whosewhose altitude altitude isis higherhigher or or equalequal to hto hi+1i+122 Contours Contours whosewhose altitude altitude isis lowerlower thanthan hhi+1i+1 but but closercloser to the to the
hierarchicalhierarchical image himage hi+1i+1 thanthan to 0to 033 Contours Contours whosewhose altitude altitude isis close to 0 close to 0
2
3
OnlyOnly the the removalremoval of the last of the last kindkind of contours of contours isis justifiedjustified
In In greygrey the contours the contours whichwhich are are removedremoved by the by the waterfallwaterfalltransformationtransformation
WATERFALLS SHORTSIGHTNESSWATERFALLS SHORTSIGHTNESS
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 86
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 87
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 88
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 89
bull A new A new hierarchyhierarchy levellevel appearsappears onlyonly if if atat least one contour least one contour isisremovedremovedbullbull The last The last levellevel isis NEVER NEVER emptyempty It Itrsquorsquos a selfs a self--blockingblocking processprocess
4 4 levelslevels of of hierarchyhierarchy onlyonlyhelliphellipThis This resultresult isis due to the due to the factfact thatthat the the watershedwatershed isis not an not an homotopichomotopic transformtransform
PROPERTIES AND EXAMPLEPROPERTIES AND EXAMPLE
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 90
WatershedWatershed
bull Maxima Maxima whichwhich are are insideinside the the catchmentcatchment basins are not basins are not takentakenintointo accountaccount
bullbull The The islandsislands appearingappearing duringduringthe the floodingflooding processprocess are are alwaysalwayssubmergedsubmerged
SEMISEMI--HOMOTOPY OF THE WATERSHEDHOMOTOPY OF THE WATERSHED
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 91
ENHANCED WATERFALL ALGORITHMENHANCED WATERFALL ALGORITHMbull Contours Contours insideinside the the laquolaquo islandsislands raquoraquo must must bebetakentaken intointo accountaccountbullbull No No laquolaquo interferenceinterference raquoraquo betweenbetween islesisles isisallowedallowedbullbull GeodesicGeodesic dual reconstruction of dual reconstruction of segmentation ssegmentation si+2i+2 by by hierarchyhierarchy hhi+1i+1
ssii
ssi+1i+1
hhi+1i+1
ssi+2i+2
RRhhi+1i+1(s(si+2i+2))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 92
EXAMPLES OF SEGMENTATIONEXAMPLES OF SEGMENTATION
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 93
EXAMPLES OF SEGMENTATION (2)EXAMPLES OF SEGMENTATION (2)
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 94
INTRODUCTION TO PILINGSINTRODUCTION TO PILINGSbull Not the Not the samesame approachapproach as as waterfallswaterfalls but but withwith identicalidentical premisespremises))
bullbull DefinitionDefinition of a of a residualresidual transformtransform((worksworks onlyonly on on valuedvalued WTS WTS ndashndash CB and SCB have CB and SCB have samesame support) support)
AtAt the first the first stepstep ψψ11 isis the the hierarchicalhierarchical image because Wimage because W00 the the WTS WTS isis the the complementarycomplementary set of the minima of set of the minima of ψψ00
ww00 = = ψψ00WeWe definedefine a a functionfunctionξξ00 = = ψψ00 on Minon Mincc((ψψ00) ) ξξ00 = max on Min(= max on Min(ψψ00))
WeWe definedefine thenthenψψ11 = R= R
ξξ ((ψψ00))00ψψ11
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 95
RESIDUES OF PILINGSRESIDUES OF PILINGS
AtAt the the nextnext stepstep the the samesame transformation transformation isis defineddefined by by usingusingthe the functionfunction ξξ11 itselfitself defineddefined fromfrom the minima of the minima of ψψ11
WeWe cancan thenthen definedefine a first a first residueresidue rr11 as the as the differencedifference betweenbetweenψψ11 and and ψψ00
rr11 = = ψψ11 ndashndash ψψ00
This This residueresidue corresponds to the corresponds to the pikingspikings whichwhich are are necessarynecessaryto to fillfill in the minima of in the minima of ψψ00
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 96
RESIDUES OF PILINGS (2)RESIDUES OF PILINGS (2)
Transformations yTransformations yii and successive and successive residuesresidues rrii ((differentdifferent colorscolors))
ψψ00 ψψ11
ψψ22 ψψ33
ψψ44
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 97
PILING RESIDUAL TRANSFORMPILING RESIDUAL TRANSFORM
DefinitionDefinition of a of a residualresidual transformtransform by by iterationiteration
ψψii isis defineddefined fromfrom ψψii--11
ψψii = R= Rξξii--11
((ψψii--11))withwithξξii--11 = = ψψii--11 on Minon Mincc((ψψii--11) ) ξξii--11 = max on Min(= max on Min(ψψii--11))
The The residueresidue rrii isis equalequal totorrii == ψψii ndashndash ψψii--11
FinallyFinally twotwo functionsfunctions θθ and q are and q are defineddefinedθθ = sup (r= sup (rii) = sup() = sup(ψψii ndashndash ψψii--11))
iiccI iI iccIIq = q = argarg max(rmax(rii) = ) = argarg max(max(ψψii ndashndash ψψii--11))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 98
PILING RESIDUAL TRANSFORM (2)PILING RESIDUAL TRANSFORM (2)
Successive Successive stepssteps of the construction of of the construction of functionfunction θθ
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 99
PILINGS AND HIERARCHYPILINGS AND HIERARCHY
bull PilingsPilings covercover somesome contours contours whenwhen somesome othersothers are are preservedpreserved((functionfunction θθ))bullbull The The preservedpreserved contours contours remainremain in sup(win sup(w00θθ) So ) So theythey cancan bebeextractedextracted by a topby a top--hathat transformtransform
WhatWhat isis the the criterioncriterion for for selectingselecting the the preservedpreserved contourscontoursbullbull WaterfallWaterfall algorithmalgorithm (primitive one)(primitive one)
Contours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheightsare are belowbelow the maximal the maximal heightheight of the of the preservedpreserved contourscontours
bullbull PilingsPilings residuesresiduesContours Contours separatingseparating regionsregions wherewhere ((insideinside) contours ) contours heightsheights(if (if anyany) are ) are atat least least twicetwice lowerlower thanthan the minimal the minimal heightheight of of the the preservedpreserved contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 100
EXAMPLEEXAMPLE
Original image Original image ψψ0 0 θθ
q Initial contours q Initial contours θ θ contourscontours
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 101
A TEMPORARY CONCLUSIONA TEMPORARY CONCLUSIONhelliphellipIt It remainsremains a a fruitfulfruitful methodologymethodologybullbull Extension to the 3D worldExtension to the 3D worldbullbull Extension to graphsExtension to graphsbullbull New New developmentsdevelopments ((residuesresidues))
TheseThese toolstools are (are (quitequite) ) easilyeasily understandableunderstandable and and handyhandybullbull There There isis usuallyusually no change of no change of spacespace (image (image spacespace))bullbull The The toolboxtoolbox isis enrichedenriched withwith affordableaffordable transformationstransformations
The performances are The performances are improvingimproving in a in a dramaticdramatic waywaybullbull New New algorithmsalgorithms especiallyespecially for for hierarchicalhierarchical segmentationsegmentationbullbull Speed of Speed of algorithmsalgorithms allowingallowing realreal--time computationtime computation
New New toolstools are are availableavailable
StrongStrong connexions connexions withwith perceptive perceptive principlesprinciples (Gestalt (Gestalt theorytheory))
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-
ICS XII 2007 Saint Etienne copy Serge BEUCHER ENSMP 102
bull S BEUCHER C LANTUEJOUL Use of watersheds in contour detection International Workshop on image processing real-time edge and motion detectionestimation Rennes Sept 1979
bull FMEYER S BEUCHER Morphological segmentation Journal of Visual Communication and Image Representation ndeg 1 Vol 1 Oct 1990
bull S BEUCHER Watershed hierarchical segmentation and waterfall algorithm Proc Mathematical Morphology and its Applications to Image Processing Fontainebleau Sept 1994 Jean Serra and Pierre Soille (Eds) Kluwer Ac Publ Nld 1994 pp 69-76
bull S BEUCHER Segmentation dimages et morphologie matheacutematique Thegravese de Doctorat Ecole des Mines de Paris Cahiers du centre de Morphologie Matheacutematique Fascicule ndeg 10 Juin1990
bull SBEUCHER F MEYER The Morphological approach of segmentation the watershed transformation In Dougherty E (Editor) Mathematical Morphology in Image Processing Marcel Dekker New York 1992
httpcmmensmpfrbibliothequehtml
httpcmmensmpfr~beucherpublihtml
(Very) SHORT BIBLIOGRAPHY(Very) SHORT BIBLIOGRAPHY
For a more complete bibliography see
- SEGMENTATION TOOLS in MATHEMATICAL MORPHOLOGY
- PRELIMINARY REMARKS
- WHAT IS IT ALL ABOUT
- A MACHINE-TOOL THE WATERSHED TRANSFORM (WTS)
- THE CLASSICAL WATERSHED ALGORITHM
- THE CLASSICAL WATERSHED ALGORITHM (2)
- WATERSHED ALGORITHMS
- BIASES AND FALSITIES WITH THE WATERSHED
- BIASES AND FALSITIES WITH THE WATERSHED (2)
- BIASES AND FALSITIES WITH THE WTS (3)
- BIASES AND FALSITIES WITH THE WATERSHED (4)
- USE OF WATERSHED
- THE GRADIENT A REMINDER
- THE MARKER-CONTROLLED WATERSHED
- EXAMPLE OF MARKER-CONTROLLED WATERSHED
- MARKER-CONTROLLED WATERSHED ALGORITHMS
- GEODESIC RECONSTRUCTION
- GEODESIC RECONSTRUCTION (2)
- SWAMPING (HOMOTOPY MODIFICATION)
- POSITION OF THE MARKERS
- POSITION OF THE MARKERS (2)
- THE SEGMENTATION PARADIGM
- WHICH CRITERIA
- APPLICATIONS
- APPLICATIONS (2)
- APPLICATIONS (3)
- APPLICATIONS (4)
- APPLICATIONS (5)
- APPLICATIONS (6)
- APPLICATIONS (7)
- APPLICATIONS (8)
- APPLICATIONS (9)
- APPLICATIONS (10)
- APPLICATIONS (11)
- CRITERIA BASED ON NUMERICAL RESIDUES
- ULTIMATE OPENING GRANULOMETRIC FUNCTION
- MARKERS GENERATION
- ROCKS SEGMENTATION
- QUASI-DISTANCES
- SEGMENTATION WITH QUASI-DISTANCES
- ANOTHER EXAMPLE
- ANOTHER EXAMPLE (2)
- GRADIENT AND QUASI-DISTANCE
- DETAILED APPLICATIONS
- DETAILED APPLICATIONS (2)
- DETAILED APPLICATIONS (3)
- DETAILED APPLICATIONS (4)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (5)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- DETAILED APPLICATIONS (6)
- HIERARCHICAL SEGMENTATION WATERFALLS
- VALUED WATERSHED
- MOSAIC IMAGE AND ITS GRADIENT
- OVER-SEGMENTATION AND PERCEPTION OF IMAGES
- GRAPH DEFINITION
- GRAPH DEFINITION AND ASSOCIATED WATERSHED
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (2)
- GRAPH DEFINITION AND ASSOCIATED WATERSHED (3)
- FROM A 3D to A PLANAR GRAPH
- IMAGE REPRESENTATION
- FIRST OVERFLOW ZONES (FOZ)
- WATERFALLS
- WATERFALLS (2)
- SIGNIFICANT CBARCS AND RECONSTRUCTION
- RECONSTRUCTION AND HIERARCHICAL IMAGE
- WATERFALLS AND MOSAIC IMAGES
- HIERARCHICAL SEGMENTATION EXAMPLE
- USE OF WATERFALLS
- USE OF WATERFALLS (2)
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- PROBLEMS WITH THE WATERFALLS
- WATERFALLS SHORTSIGHTNESS
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- PROPERTIES AND EXAMPLE
- SEMI-HOMOTOPY OF THE WATERSHED
- ENHANCED WATERFALL ALGORITHM
- EXAMPLES OF SEGMENTATION
- EXAMPLES OF SEGMENTATION (2)
- INTRODUCTION TO PILINGS
- RESIDUES OF PILINGS
- RESIDUES OF PILINGS (2)
- PILING RESIDUAL TRANSFORM
- PILING RESIDUAL TRANSFORM (2)
- PILINGS AND HIERARCHY
- EXAMPLE
- A TEMPORARY CONCLUSIONhellip
- (Very) SHORT BIBLIOGRAPHY
-