1 digital topology and cross-section topology for grayscale image processing m. couprie a 2 si lab.,...
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Digital topology andcross-section topology for grayscale image processing
M. Couprie A2SI lab., ESIEEMarne-la-Vallée, Francewww.esiee.fr/a2si
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What is topology ?
« A topologist is interested in those properties of a thing that, while they are in a sense geometrical, are the most permanent- the ones that will survive distortion and stretching. »
Stephen Barr,"Experiments in Topology",1964
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Aims of this talk
Present the main notions of digital topologyShow their usefulness in image processing
applicationsStudy some topological properties of digital
grayscale images (ie. functions from Zn to Z)Topology-preserving image transformsTopology-altering image transformsApplications to image processing (filtering,
segmentation…)
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Some milestones
J.C. Maxwell (1870): ‘On hills and dales ’A. Rosenfeld (1970’s): digital topologyA. Rosenfeld (1980’s): fuzzy digital
topologyS. Beucher (1990): definition of
homotopy between grayscale imagesG. Bertrand (1995): cross-section
topology
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Contributors
Gilles BertrandMichel CouprieLaurent PerrotonZouina Aktouf (Ph.D student)Francisco Nivando Bezerra (Ph.D
student) Xavier Daragon (Ph.D student)Petr Dokládal (Ph.D student)Jean-Christophe Everat (Ph.D student)
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Outline of the talk
Digital topology for binary imagesApplications to graylevel image
processing
Cross-section topology for grayscale images
Topology-altering transforms, applications
Part 1
Part 2
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Digital topology for binary images
8-adjacency, 4-adjacency
in black: X (object)
in white: X (background)
.
path, connected components
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Digital topology for binary images
8-adjacency, 4-adjacency
in black: X (object)
in white: X (background)
.
path, connected components
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8-adjacency, 4-adjacency
in black: X (object)
in white: X (background)
.
path, connected components
Digital topology for binary images
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Digital topology for binary images
8-adjacency, 4-adjacency
in black: X (object)
in white: X (background)
.
path, connected components
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More examples
Black:4, white:8 Black:8, white:4
3 1 2 2
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Jordan property
Any simple closed curve divides the plane into 2 connected components.
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Jordan property (cont.)
A subset X of Z2 is a simple closed curve if each point x of X has exactly two neighbors in X
4-curve 8-curve
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Jordan property (cont.)
A subset X of Z2 is a simple closed curve if each point x of X has exactly two neighbors in X
Not a 4-curve
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Jordan property (cont.)
A subset X of Z2 is a simple closed curve if each point x of X has exactly two neighbors in X
Not an 8-curve
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Jordan property (cont.)
The Jordan property does no hold if X and its complement have the same adjacency
8-adjacency 4-adjacency
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Topology preservation - notion of simple point
A topology-preserving transform must preserve the number of connected components of both X and X.
Definition (2D): A point p is simple (for X) if its modification (addition to X, withdrawal from X) does not change the number of connected components of X and X.
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Simple point: examples
Set X (black points)
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Simple point: examples
Simple point of X
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Simple point: examples
Simple point of X
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Simple point: counter-examples
Non-simple point (background component creation if deleted)
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Simple point: counter-examples
Non-simple point (component splitting if deleted)
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Simple point: parallel deletion
Deleting simple points in parallel may change the topology
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Simple point: local characterization ?
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Simple point: local characterization ?
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Connectivity numbers
T(p)=number of Connected Components of (X - {p}) N(p) where N(p)=8-neighborhood of p
T(p)=number of Connected Components of (X - {p}) N(p)A
Characterization of simple points (local): p is simple iff T(p) = 1 and T(p) = 1 A
T=2,T=2 T=1,T=1 T=1,T=0 T=0,T=1
U
U
Interior point Isolated point
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Skeletons
We say that Y is a skeleton of X if Y may be obtained from X by sequential deletion of simple points
If Y is a skeleton of X and if Y contains no simple point, then we say that Y is an ultimate skeleton of X
If Y is a skeleton of X and if Y contains only non-simple points and ‘end points’, then we say that Y is a curve skeleton of X
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End points
We say that p is an end point for X if (X - {p}) N(p) contains exactly one point
End point End point (8) Non-end point Non-end point
U
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Skeletons (examples)
Original image Curve skeleton Ultimate skeleton
End points
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Homotopy
We say that X and Y are homotopic (i.e. they have the same topology) if Y may be obtained from X by sequential addition or deletion of simple points
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Basic thinning algorithms
Ultimate thinning(X)Repeat until stability:
Select a simple point p of XRemove p from X
Curve thinning(X)Repeat until stability:
Select a simple, non-end point p of XRemove p from X
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Centering skeletons
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Breadth-first thinning
Simple pointsdetectedduring the 1stiterationCandidatesfor the 2nditeration
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Breadth-first thinning
Breadth-first ultimate thinning(X)T := X ; T’ := Repeat until stability
While p T do T := T \ {p} If p is a simple point for X then
X := X \ {p} For all q neighbour of p, q X,
do T’ := T’ {q}
T := T’ ; T’ :=
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Directional thinning
(only in 2D)
NorthSouthEastWest
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Parallel directional algorithm
In 2D, simple and non-end points of the same type (N,S,E,W) can be removed in parallel
Directional Curve thinning(X) Repeat until stability:
For dir =N,S,E,W Let Y be the set of all simple, non-end points of X of type dir X = X \ Y
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Does not work in 3D
In 3D, there are 6 principal directions: N,S,E,W,Up,Down
Up
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Thinning guided by a distance map
Let X be a subset of Z2, the distance map DMX is defined by, for all point p:
DMX(p) = min{dist(p,q), q not in X}
where dist is a distance (e.g. city block, chessboard, Euclidean)
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Discrete distances
City block distance (d4)d4(p,q) = length of a shortest 4-
pathbetween p and q
Chessboard distance (d8)d8(p,q) = length of a shortest 8-
pathbetween p and q
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Discrete distance maps
1 11 1 1 111111111
11
11111111
11 1 11 1
1 111
11
22222222
22222222
2222
22 22
222233
3 3 3 3
3 3 3 333
33
44
44
22222222
2 2222222
2222
22 22
33 3
3
3 3 3 333
33
4 4
1 11 1 1 111111111
11
11111111
11 1 11 1
1 111
11
1 1
11
11
11
d4 d8
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Discrete and Euclidean distance maps
d8
d4de
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Thinning guided by a distance map: algorithm
Ultimate guided thinning(X)Compute the distance map DMX
Repeat until stability:Select a simple point p of X
such that DMX(p) is minimalRemove p from X
Curve guided thinning(X)Id., replace « simple » by « simple non end »
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Thinning guided by a distance map: results
d8
d4de
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The 3D case
A topology-preserving transform must preserve:
- number of connected components of X
- number of connected components of X
- number of « holes » (or « tunnels »)
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3D hole (tunnel)
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3D hole (tunnel)
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3D-skeletons
Originalobject
Surfaceskeleton
Curveskeleton
Ultimateskeleton
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Topological classification of points (G. Bertrand)
T=1,T=1 T=1,T=2 T=3,T=1(1D isthmus)(simple point) (2D isthmus)
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3D hole closing
G. BertrandZ. Aktouf (Ph. D)L. Perroton
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3D hole closing
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3D hole closing
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3D hole closing
T=2,T=1 (1D isthmus)
T=1,T=2 (2D isthmus)
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3D hole closing
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3D hole closing algorithm
Let X be an object and Y be a simply connected set containing X
We remove iteratively all points x of Y \ X such that T(x) = 1(i.e. simple points and 1D isthmus)
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3D hole closing
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3D hole closing using a distance map
The points are processed in an order depending on their distance from X (points far from X are selected first)
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3D hole closing using a distance map
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3D hole closing using a distance map
When a point p with T=2 is selected, its removal would create a hole. The value of the distance map for this point (DMX(p)) gives an information on the « size » of the corresponding hole in X.
We can decide to let a hole open or to close it according to this criterion.
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3D hole closing using a distance map
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3D hole closing: illustration
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3D hole closing: illustration
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3D hole closing: illustration
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Outline of the talk
Digital topology for binary imagesApplications to graylevel image
processing
Cross-section topology for grayscale images
Topology-altering transforms, applications
Part 1
Part 2
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Liver vascular system segmentation
Contrast-enhanced tomography scan of liver
(2-D cut from a 3-D image)
G. BertrandP. Dokladal (Ph. D)
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Intensity-guided skeletonization
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Segmentation results
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Skeleton Filtering (1)
Noisy image Noisy skeleton
Identification of relevant
components
Filtered skeleton
Filtering criterion based on minimum accepted meanluminosity of each skeleton segment.
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Skeleton Filtering (2)
Two criteria based on local information:
•mean luminosity
•amplitude difference
Filtering criterion based on detection of irrelevant end points
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Skeleton Filtering (results)
Non filtered Filtered
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Segmentation of the brain cortex
M. CouprieX. Daragon (PhD)
Basic assumption: the geometry of the brain is complex, but its topology is simple
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Segmentation of the brain cortex
Cortex (gray matter)
White matter
Cerebro-spinal fluid
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Segmentation of the brain cortex: main steps
Pre-segmentation
Extension to the cortex
White matter extraction
Cortex extraction
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Extraction of the white matter
Morphological closing with topological control
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Morphological closing with topological control
A: original image B: opening of Awith a small disc
C: ultimate skeletonof A in B
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Extension to the cortex
W.M.
W.M.
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Extension to the cortex: use of a distance map
Section of the 3D segmentation of the white matter
Section of its 3D distance map (limited to distances < 1cm)
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Constraint set: centers of maximal balls
Distance map (section)
Centers of maximal balls
A maximal ball (for a set X) is a ballincluded in X which is not included
in any other ball included in X.The set of the centers of maximal balls
for X can be computed fromthe distance map.
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Results (sections of 3D images)
Without constraint set With constraint set
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Results (3D view)
Without constraint set With constraint set
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Results (3D view)
White matter Cortex
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Conclusion (1st part)
Image simplification with topology preservation
Controled topology modificationBinary topological operators guided
by a distance map or by a grayscale image
Efficient implementations
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Outline of the talk
Digital topology for binary imagesApplications to graylevel image
processing
Cross-section topology for grayscale images
Topology-altering transforms, applications
Part 1
Part 2
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Cross-section topology
G. BertrandM. CouprieJ.C. Everat (PhD)F.N. Bezerra (PhD)
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Cross-section topology
Basic idea: consider the topology of each cross-section (threshold) of a function.
Given a function F (Z2 Z) and k in Z, we define the cross-section Fk as the set of points p of Z2 such that F(p) k.
We say that two functions F and G are homotopic if, for every k in Z, Fk and Gk are homotopic (in the binary sense)
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Homotopy: an illustration
x
yF(x,y)
F3
F2
F1
x
G(x,y)
G3
G2
G1
y
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Destructible point(G. Bertrand)
Definition: a point p is destructible (for F) if it is simple for Fk, with k = F(p) .
Property: p is destructible iff its value may be lowered by one without changing the topology of any cross-section.
Definition: a point p is constructible (for F) if it is destructible for -F (duality)
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Destructible point: examples
x
yF(x,y)
F3
F2
F1
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Destructible point: examples
x
yF(x,y)
F3
F2
F1
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Destructible point: counter-examples
x
yF(x,y)
F3
F2
F1
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Destructible point: counter-examples
x
yF(x,y)
F3
F2
F1
Componentdeleted
Componentsplitted
Backgroundcomponentcreated
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Connectivity numbers
N+(p) = {q in N(p), F(q) F(p)}T+(p) = number of Conn. Comp. of N+(p)
.
N--(p) = {q in N(p), F(q) < F(p)}T--(p) = number of Conn. Comp. of N--(p)
.
N++, T++, N-, T- : similar If an adjacency relation (eg. 4) is chosen
for T+, T++, then the other adjacency (8) must be used for T-, T--
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Connectivity numbers: examples
1 2 19 5 19 9 9
T+ = 1T-- = 1
1 2 89 5 89 1 1
T+ = 2T-- = 2
1 2 11 5 12 2 1
T+ = 0T-- = 1
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Destructible point: local characterization
Property: the point p is destructible iff T+(p) = 1 and T--(p) = 1
1 2 19 5 19 9 9
T+ = 1T-- = 1
1 2 89 5 89 1 1
T+ = 2T-- = 2
1 2 11 5 12 2 1
T+ = 0T-- = 1
destructible non-destructible non-destructible
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Classification of points(G. Bertrand)
The local configuration of a point p corresponds to exactly one of the eleven following cases:
well (T- = 0) minimal constructible
(T++ = T- = 1, T-- = 0) minimal convergent
(T++ > 1, T-- = 0) constructible divergent
(T++ = T- = 1, T-- > 1)
peak (T+ = 0) maximal destructible
(T+ = T-- = 1, T++ = 0) maximal divergent
(T-- > 1, T++ = 0) destructible convergent
(T+ = T-- = 1, T++ > 1) interior (T++ = T-- = 0) simple side (T+ = T-- = T++ = T- =
1) saddle (T++ > 1, T-- > 1)
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Classification of points: examples
1 2 19 5 19 9 9
1 2 89 5 89 1 1
1 2 11 5 12 2 1
Simple side Saddle
Peak
1 2 19 9 19 9
1 29 9 9
1 1
1 2 15
Maximaldestructible
Maximaldivergent
Destructibleconvergent
5 5 55 5 55 5 5
Interior
1
2
199
995
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Grayscale skeletons
We say that G is a skeleton of F if G may be obtained from F by sequential lowering of destructible points. .
If G is a skeleton of F and if G contains no destructible point, then we say that G is an ultimate skeleton of F
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Ultimate skeleton: 1D example
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Ultimate skeleton: illustration
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Ultimate skeleton: illustration
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Ultimate skeleton: 1D example
Regional minima
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Ultimate skeleton: 2D example
Original image F
Regionalminimaof F(white)
UltimateskeletonG of F
Regionalminimaof G(white)
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(
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Watersheds
Powerful segmentation operator from the field of Mathematical Morphology
Introduced as a tool for segmenting grayscale images by S. Beucher and C. Lantuejoul in the 70s
Efficient algorithms based on immersion simulation were proposed by L. Vincent, F. Meyer, P. Soille (and others) in the 90s
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Watersheds
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Watersheds: illustration
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)
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Ultimate skeleton vs. Watershed line
1 1 111 3
1 1 1
31 311
3 3
333
3
3 3 3 3
3
32
1 1 1
111 1 1 1 1 1 1 1 1
1111111
Ultimate skeleton(non-minimal points)
1 1 11 6 61 6 3
1 1 16 63
1 6 31 61
73 3
37 7
777 7 73
3
3
3 3 3 3
3
3
6 6 6
2 5666 6 6 6 6 6 6
66666
1 1 1
111 1 1 1 1 1 1 1 1
1111111
Watershed
3
3
3
3
1
1
3
3
3
3
1
1
3
3
3
3
6
5
6
1
1
3
3
3
3
6
5
6
1
1
6 66
6 6
66
7 7 777
7 7 7
6 6 6
666 6 6 6 6 6 6
66666
5
6
6
5
6
6
5
6 66
6 6
66
7 7 777
7 7 7
6 6 6
666 6 6 6 6 6 6
66666
5
6
6
5
6
6
5
6 66
6 6
66
7 7 777
7 7 7
6 6 6
666 6 6 6 6 6 6
66666
6
6
6
6
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Ultimate skeleton vs. Watershed line
1 1 11 6 61 6 1
1 1 16 61
1 6 11 61
71 1
17 7
777 7 71
1
1
1 1 1 1
1
1
6 6 6
2 6666 6 6 6 6 6 6
66666
1 1 1
111 1 1 1 1 1 1 1 1
1111111
1
1
1
1
6
6
6
1
1
1
1
1
1
6
6
6
1
1
6 66
6 6
66
7 7 777
7 7 7
6 6 6
666 6 6 6 6 6 6
66666
6
6
6
6A watershed point cannotbe characterized locally
Central point: belongsto the watershed line
Central point: does notbelong to the watershed line
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Thinness
In the previous examples, the set of non-minimal points of an ultimate skeleton was « thin » (a set X is thin if it contains no interior point). Is it always true ?
The answer is no, as shown by the following counter-examples.
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Thinness
11
31
3 3 3 3 3 3 3 31 1
111 1 1
13 3 3 3 3 3 3 3 3
3333 3
33
33
3 133
333
111 3 33 13 31 1 1 133 33 3
1 1 1 133 3 3 33 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 33333 3
33
33
333
333
3 333 333 33 333 3 3 3
3 3 3 3 3 3 3 3 3
3
1 1 1 1 11 11 1 1 1 1 1
1 1 11 1 11 1 11 1 1
1 1 11 1 11 1 11 1 1
3 3 3 3 3 33
333
333
3
333
333
3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 3
322222
3 3 3 3 3 33
333
333
3
333
333
3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 3
322222
2
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111
2 22
22 2
Thinness
1 1 1 1 11 11 1 1
1 1 11 1 11 11 1 1
1 1 1
1 1 11
11 1 1 1
1 1 1 1 11 1 1 1 111
1 1 11 1 1
11
1 1 11 1 11 1 11 1 11 1 11 1 1
1 11 1 1
1 11 1 1
3 3 3 3 3 3 3 3 33 3 3 3 3
3 3 3
33
333
33 3
33 3
32 2
2 2 22 2 22 2 2
222
3
333
33333 3 3 3 3 3 3 3 33 3 3 3
3
333
33333
3 3 3 3 3
3
3
3
333
133
333
3 3 3 3
22
22
221
11
3 3 3 3 3
3
3
3
333
33
333
3 3 3 3
2 2 22 22
22 2
2 2 2
2 22
22 2
3 3 3 3 3 3 3 3 33 3 3 3 3
3 3 3
33
333
33 3
33 3
3
2 2 22 2 22 2 2
222
3
333
33333 3 3 3 3 3 3 3 33 3 3 3
3
333
33333
2 2 22 2 22 2 2
2
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Basic algorithm
Basic ultimate grayscale thinning(F)
Repeat until stability: Select a destructible point p for F F(p) := F(p) – 1
Inefficient: O(n.g), where: n is the number of pixels g is the maximum gray level
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Lowest is best
The central point is destructible: it can thus be lowered down to 5 without changing the topology.
It can obviously be lowered more:- down to 3 (since there is no value
between 6 and 3 in the neighborhood)- down to 1 (we can check that once at
level 3, the point is still destructible)
3 1 19 6 19 9 9
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Two special values
If p is destructible, we define:
-(p)=highest value strictly lower than F(p) in the neighborhood of p
-(p)=lowest value down to which F(p) can be lowered without changing the topology
3 1 19 6 19 9 9
Here: -(p)=3
-(p)=1
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Better but not yet good
If we replace: F(p) := F(p) – 1 in the basic
algorithm by: F(p) := -(p), we get a faster algorithm. But its complexity is still bad. Let us show why:
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Fast algorithm
Fast ultimate grayscale thinning(F)
Repeat until stability: Select a destructible point p for F of minimal graylevel
F(p) := -(p)
- Can be efficiently implemented thanks to a hierarchical queue- Execution time roughly proportional to n (number of pixels)
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Complexity analysis: open problem
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Outline of the talk
Digital topology for binary imagesApplications to graylevel image
processing
Cross-section topology for grayscale images
Topology-altering transforms, applications
Part 1
Part 2
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Topology-altering transforms, applications
G. BertrandM. CouprieJ.C. Everat (PhD)F.N. Bezerra (PhD)
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Altering the topology
Control over topology modification.
Criteria: Local contrast : notion of -skeleton Regional contrast : regularization Size : topological filtering Topology : crest restoration
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Non-homotopic operators
Topology preservation: strong restriction
Our goal: Change topology in a controlled way
over segmentation
regional minima
segmented regions
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-destructible point
-destructible
not -destructible
Illustration (1D profile of a 2D image)
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-destructible point
Definition:
Let X be a set of points, we define F-(X)=min{F(p), p
in X}
Let be a positive integer
A destructible point p is -destructible A k-divergent point p is -destructible if at least k-1
connected components ci (i=1,…,k-1) of N--(p) are such that F(p) - F-(ci)
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-skeleton
.
We say that G is a -skeleton of F if G may be obtained from F by sequential lowering of -destructible or peak points
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-skeleton : example
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1 2 19 9 11 9 21 9 19 9 11 1 1
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Original image
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1 1 12 9 91 9 1
1 2 19 9 11 9 21 9 19 9 11 1 1
2 9 11 9 91 1 2-skeleton (=3)
1 21 11 11 11 11 1
The order of operations matters
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-skeleton : examples
= 0 = 15 = 30
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Leveled skeleton
We say that G is a leveled skeleton of F if G may be obtained from F by sequential lowering of destructible or peak points.
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Leveled skeleton: example
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2 8 11 9 91 1 2
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2 8 11 8 91 1 2
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Original image After 7 steps After 11 steps
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Leveled skeleton: example
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1 2 16 6 11 7 21 7 18 7 11 1 1
2 8 11 8 81 1 2
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1 2 16 6 11 6 21 6 16 6 11 1 1
2 6 11 6 81 1 2
1 1 12 6 61 6 1
1 2 16 6 11 6 21 6 16 6 11 1 1
2 6 11 6 61 1 2After 18 steps(final result)
After 11 steps(reminder)
After 17 steps
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Leveled skeleton: example
Original image
Regional minima Regional minima
Leveled skeleton
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Regularization
In a leveled skeleton, the regional minima are separated by ‘thin crest lines’ .
The graylevel on these lines correspond to the ‘altitude of the lowest pass connecting two neighboring minima’ (in the skeleton, and in the original image as well) .
This allows to detect and modify ‘irregular crest points’
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Regularization (cont.)
a
b
x
If a > b then x is irregular
a
bx
If a > b then x is irregular
Leveled skeleton
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Regularization: example
Original image Regularized leveled skeleton
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Binary reconstruction
Original image Regularized skeleton
a
b
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Regularization and reconstruction: segmentation without any parameter
Original image Regularized leveled skeleton
Binaryreconstruction
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Topological filtering
A: original C: reconstruction ofB under A
B: thinning+peak deletion
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Topological filtering (cont.)
Original image
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Topological filtering (cont.)
Homotopic thinning (n steps)
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Topological filtering (cont.)
Peak deletion
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Topological filtering (cont.)
Homotopic reconstruction
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Topological filtering (cont.)
Final result
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Comparison with the morphological approach
Morphological opening (erosion,dilation): makes no difference between a disk of diameter d and an elongated object of thickness d.
Area opening: makes no difference between a disk and an elongated object having the same area
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Crest restoration
Motivation
Thinning + thresholding
GradientOriginal image
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Crest restoration (cont.)
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p is a separating point if there is k such that T(p, Fk)=2
p is extensible if p is a separating point, and
p is a constructible or saddle point, and
there is a point q in its neighborhood that is an end point or an isolated point for Fk, with k=F(p)+1
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Crest restoration (cont.)
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p is a separating point if there is k such that T(p, Fk)=2
p is extensible if p is a separating point, and
p is a constructible or saddle point, and
there is a point q in its neighborhood that is an end point or an isolated point for Fk, with k=F(p)+1
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Crest restoration (cont.)
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Crest restoration (cont.)
Thinning +thresholding
Thinning +crest restoration +
thresholdingGradient
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Crest restoration (cont.)
Thinning +crest restoration +
thresholding(1 parameter)
Thinning+hysteresis thresholding
(2 parameters)
Thinning+hysteresis thresholding
(2 parameters)
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Crest Restoration: result
‘Significant’ crests have been highlighted (in green)
Before crest restoration:
After crest restoration:
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Centering grayscale skeletons
Breadth-first ultimate thinning(F)T := domain of F ; T’ := Repeat until stability
While p T do T := T \ {p} If p is a destructible point for F then
F(p) := -(p) For all q neighbour of p, do T’ := T’ {q}
T := T’ ; T’ :=
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Breadth-first ultimate thinning : example
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New strategies to reduce anisotropy
Generalized Euclidean distance map(computes one distance map for each cross-section)
Dynamic estimation of Euclidean distance(uses the Danielson’s principle and approximation)
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Reducing anisotropy: results
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Reducing anisotropy: results
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Reducing anisotropy: results
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Conclusion (2nd part)
Strict preservation of both topological and grayscale information
Combining topology-preserving and topology-altering operators
Control based on several criteria (contrast, size, topology)
Strategies to reduce anisotropy
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Perspectives
Study of complexityExtension to 3DTopology in orders (G. Bertrand)
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Perspectives
G. Bertrand, J. C. Everat and M. Couprie: "Image segmentation through operators based upon topology", Journal of Electronic Imaging, Vol. 6, No. 4, pp. 395-405, 1997.
M. Couprie, F.N. Bezerra, Gilles Bertrand: "Topological operators for grayscale image processing", Journal of Electronic Imaging, Vol. 10, No. 4, pp. 1003-1015, 2001.
www.esiee.fr/~coupriem/Sdi/publis.html