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Digital Camera and Computer Vision LaboratoryDepartment of Computer Science and Information Engineering
National Taiwan University, Taipei, Taiwan, R.O.C.
Computer and Robot Vision I
Chapter 5 Mathematical MorphologyPresented by: 傅楸善 & 王林農
[email protected]指導教授 : 傅楸善 博士
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5.1 Introduction
mathematical morphology works on shape shape: prime carrier of information in machine
vision morphological operations: simplify image data,
preserve essential shape characteristics, eliminateirrelevancies
shape: correlates directly with decomposition ofobject, object features, object surface defects, assembly defects
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5.2 Binary Morphology
set theory: language of binary mathematical morphology
sets in mathematical morphology: represent shapes
Euclidean N-space: EN
discrete Euclidean N-space: ZN
N=2: hexagonal grid, square grid
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5.2 Binary Morphology (cont’)
dilation, erosion: primary morphological operations
opening, closing: composed from dilation, erosion
opening, closing: related to shape representation, decomposition, primitive extraction
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5.2.1 Binary Dilation
dilation: combines two sets by vector addition of set elements
dilation of A by B:
addition commutative dilation commutative:
binary dilation: Minkowski addition
BA
ABBA
} and somefor |{ BbAabacEcB A N
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5.2.1 Binary Dilation (cont’)
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5.2.1 Binary Dilation (cont’)
A: referred as set, image B: structuring element: kernel dilation by disk: isotropic swelling or expansion
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5.2.1 Binary Dilation (cont’)
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5.2.1 Binary Dilation (cont’)
dilation by kernel without origin: might not have common pixels with A
translation of dilation: always can contain A
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5.2.1 Binary Dilation (cont’)
=lena.bin.128=
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5.2.1 Binary Dilation (cont’)
=lena.bin.dil= By structuring
element :
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5.2.1 Binary Dilation (cont’)
N4: set of four 4-neighbors of (0,0) but not (0,0,) 4-isolated pixels removed only points in I with at least one of its 4-neighbors r
emain At: translation of A by the point t
removal noisefor )( N IJJ 4 I
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5.2.1 Binary Dilation (cont’)
dilation: union of translates of kernel
addition associative dilation associative
associativity of dilation: chain rule: iterative rule dilation of translated kernel: translation of dilation
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5.2.1 Binary Dilation (cont’)
dilation distributes over union
dilating by union of two sets: the union of the dilation
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5.2.1 Binary Dilation (cont’)
dilating A by kernel with origin guaranteed to contain A
extensive: operators whose output contains input dilation extensive when kernel contains origin
dilation preserves order
increasing: preserves order
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5.2.2 Binary Erosion
erosion: morphological dual of dilation erosion of A by B: set of all x s.t.
erosion: shrink: reduce:
BbAb x every for
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5.2.2 Binary Erosion (cont’)
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5.2.2 Binary Erosion (cont’)
=Lena.bin.ero=
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5.2.2 Binary Erosion (cont’)
erosion of A by B: set of all x for which B translated to x contained in A
if B translated to x contained in A then x in A B erosion: difference of elements a and b
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5.2.2 Binary Erosion (cont’)
dilation: union of translates erosion: intersection of negative translates
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5.2.2 Binary Erosion (cont’)
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5.2.2 Binary Erosion (cont’)
Minkowski subtraction: close relative to erosion
Minkowski subtraction: erosion: shrinking of the original image antiextensive: operated set contained in the o
riginal set erosion antiextensive: if origin contained in ke
rnel
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5.2.2 Binary Erosion (cont’)
if then because
eroding A by kernel without origin can have nothing in common with A
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5.2.2 Binary Erosion (cont’)
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5.2.2 Binary Erosion (cont’)
•dilating translated set results in a translated dilation
•eroding by translated kernel results in negatively translated erosion
•dilation, erosion: increasing
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5.2.2 Binary Erosion (cont’)
eroding by larger kernel produces smaller result
Dilation, erosion similar that one does to foreground, the other to background
similarity: duality dual: negation of one equals to the other on negated
variables DeMorgan’s law: duality between set union and inter
section
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5.2.2 Binary Erosion (cont’)
negation of a set: complement
negation of a set in two possible ways in morphology logical sense: set complement geometric sense: reflection: reversing of set orientation
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5.2.2 Binary Erosion (cont’)
complement of erosion: dilation of the complement by reflection
Theorem 5.1: Erosion Dilation Duality
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5.2.2 Binary Erosion (cont’)
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5.2.2 Binary Erosion (cont’)
Corollary 5.1: erosion of intersection of two sets: intersectio
n of erosions
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5.2.2 Binary Erosion (cont’)
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5.2.2 Binary Erosion (cont’)
erosion of a kernel of union of two sets: intersection of erosions
erosion of kernel of intersection of two sets: contains union of erosions
no stronger
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5.2.2 Binary Erosion (cont’)
chain rule for erosion holds when kernel decomposable through dilation
duality does not imply cancellation on morphological equalities
containment relationship holds
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5.2.2 Binary Erosion (cont’)
genus g(I): number of connected components minus number of holes of I 4-connected for object, 8-connected for background
8-connected for object, 4-connected for background
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5.2.2 Binary Erosion (cont’)
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5.2.2 Binary Erosion (cont’)
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Joke
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5.2.3 Hit-and-Miss Transform
hit-and-miss: selects corner points, isolated points, border points
hit-and-miss: performs template matching, thinning, thickening, centering
hit-and-miss: intersection of erosions J,K kernels satisfy hit-and-miss of set A by (J,K)
hit-and-miss: to find upper right-hand corner
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5.2.3 Hit-and-Miss Transform (cont’)
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5.2.3 Hit-and-Miss Transform (cont’)
J locates all pixels with south, west neighbors part of A
K locates all pixels of Ac with south, west neighbors in Ac
J and K displaced from one another Hit-and-miss: locate particular spatial pattern
s
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5.2.3 Hit-and-Miss Transform (cont’)
hit-and-miss: to compute genus of a binary image
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5.2.3 Hit-and-Miss Transform (cont’)
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5.2.3 Hit-and-Miss Transform (cont’)
hit-and-miss: thickening and thinning hit-and-miss: counting hit-and-miss: template matching
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5.2.4 Dilation and Erosion Summary
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5.2.4 Dilation and Erosion Summary (cont’)
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5.2.5 Opening and Closing
dilation and erosions: usually employed in pairs B K: opening of image B by kernel K
B K closing of image B by kernel K
B open under K: B open w.r.t. K: B= B K B open under K: B open w.r.t. K: B= B K
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5.2.5 Opening and Closing (cont’)
=lena.bin.open=
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5.2.5 Opening and Closing (cont’)
morphological opening, closing: no relation to topologically open, closed sets
opening characterization theorem
A K: selects points covered by some translation of K, entirely contained in A
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5.2.5 Opening and Closing (cont’)
opening with disk kernel: smoothes contours, breaks narrow isthmuses
opening with disk kernel: eliminates small islands, sharp peaks, capes
closing by disk kernel; smoothes contours, fuses narrow breaks, long, thin gulfs
closing with disk kernel: eliminates small holes, fill gaps on the contours
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5.2.5 Opening and Closing (cont’)
=lena.bin.close=
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5.2.5 Opening and Closing (cont’)
unlike erosion and dilation: opening invariant to kernel translation
opening antiextensive like erosion and dilation: opening increasing
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5.2.5 Opening and Closing (cont’)
A K: those pixels covered by sweeping kernel all over inside of A
F: shape with body and handle L: small disk structuring element with radius just lar
ger than handle width extraction of the body and handle by opening and taking the residue
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5.2.5 Opening and Closing (cont’)
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5.2.5 Opening and Closing (cont’)
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5.2.5 Opening and Closing (cont’)
extraction of trunk and arms with vertical and horizontal kernels
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5.2.5 Opening and Closing (cont’)
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5.2.5 Opening and Closing (cont’)
extraction of base trunk horizontal and vertical areas
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5.2.5 Opening and Closing (cont’)
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5.2.5 Opening and Closing (cont’)
noisy background line segment removal
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5.2.5 Opening and Closing (cont’)
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5.2.5 Opening and Closing (cont’)
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5.2.5 Opening and Closing (cont’)
decomposition into parts
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5.2.5 Opening and Closing (cont’)
closing: dual of opening
like opening: closing invariant to kernel translation
closing extensive like dilation, erosion, opening: closing
increasing
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5.2.5 Opening and Closing (cont’)
opening idempotent
closing idempotent
if L K not necessarily follows that
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5.2.5 Opening and Closing (cont’)
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5.2.5 Opening and Closing (cont’)
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5.2.5 Opening and Closing (cont’)
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5.2.5 Opening and Closing (cont’)
closing may be used to detect spatial clusters of points
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Joke
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5.2.6 Morphological Shape Feature Extraction
morphological pattern spectrum: shape-size histogram [Maragos 1987]
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5.27 Fast Dilations and Erosions
decompose kernels to make dilations and erosions fast
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5.3 Connectivity
morphology and connectivity: close relation
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5.3.1 Separation Relation
S separation if and only if S symmetric, exclusive, hereditary, extensive
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5.3.2 Morphological Noise Cleaning and Connectivity
images perturbed by noise can be morphologically filtered to remove some noise
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5.3.3 Openings Holes and Connectivity
opening can create holes in a connected set that is being opened
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5.3.4 Conditional Dilation
select connected components of image that have nonempty erosion conditional dilation J ,
defined iteratively J0 = J J are points in the regions we want to select
conditional dilation J =Jm
where m is the smallest index Jm=Jm-1
DI|
DI|
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5.4 Generalized Openings and Closings
generalized opening: any increasing, antiextensive, idempotent operation
generalized closing: any increasing. extensive, idempotent operation
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Joke
End
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Hit and Miss (cont’)
hit-and-miss: thickening and thinning
hit-and-miss: template matching
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DC & CV Lab.DC & CV Lab.CSIE NTU
Hit and Miss (cont’)
hit-and-miss: thickening
)),(...)},(),((...{[ 8822111 KJKJKJAA nn
),(),( KJAAKJA
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Hit and Miss (cont’)
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Hit and Miss (cont’)
hit-and-miss: thinning
)),(...)},(),((...{[ 8822111 KJKJKJAA nn
),(),( KJAAKJA
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Hit and Miss (cont’)
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DC & CV Lab.DC & CV Lab.CSIE NTU
Hit and Miss (cont’)
hit-and-miss: template matching
cxx ITWandIT )(
})()(|{ cx
cxx IWKTandIKTx
)](,[ KTWKTI
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Hit and Miss (cont’)
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5.5 Gray Scale Morphology
binary dilation, erosion, opening, closing naturally extended to gray scale
extension: uses min or max operation gray scale dilation: surface of dilation of
umbra gray scale dilation: maximum and a set of
addition operations gray scale erosion: minimum and a set of
subtraction operations
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5.5.1Gray Scale Dilation and Erosion
top: top surface of A: denoted by
umbra of f: denoted by
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5.5.1Gray Scale Dilation and Erosion (cont’)
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5.5.1Gray Scale Dilation and Erosion (cont’)
gray scale dilation: surface of dilation of umbras
dilation of f by k: denoted by
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5.5.1Gray Scale Dilation and Erosion (cont’)
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5.5.1Gray Scale Dilation and Erosion (cont’)
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5.5.1Gray Scale Dilation and Erosion (cont’)
-36
0
36
-40
-30
-20
-10
0
10
20
30
40
-1.5 -1 -0.5 0 0.5 1 1.5
K
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5.5.1Gray Scale Dilation and Erosion (cont’)
1923
9
29
-12-15
-10
-5
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6 7 8 9
F
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5.5.1Gray Scale Dilation and Erosion (cont’)
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5.5.1Gray Scale Dilation and Erosion (cont’)
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5.5.1Gray Scale Dilation and Erosion (cont’)
-17
19
55 59
45
65
24
-30
-20
-10
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6 7 8 9 10
f dilation by k
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5.5.1Gray Scale Dilation and Erosion (cont’)
gray scale erosion: surface of binary erosions of one umbra by the other umbra
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5.5.1Gray Scale Dilation and Erosion (cont’)
=lena.im=
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5.5.1Gray Scale Dilation and Erosion (cont’)
=lena.im.dil=
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DC & CV Lab.DC & CV Lab.CSIE NTU
5.5.1Gray Scale Dilation and Erosion (cont’)
Structuring Elements:
Value=0
* * ** * * * ** * * * ** * * * *
* * *
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5.5.1Gray Scale Dilation and Erosion (cont’)
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5.5.1Gray Scale Dilation and Erosion (cont’)
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5.5.1Gray Scale Dilation and Erosion (cont’)
-36
0
36
-40-35-30-25-20-15-10-505
10152025303540
-1.5 -1 -0.5 0 0.5 1 1.5
K
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DC & CV Lab.DC & CV Lab.CSIE NTU
5.5.1Gray Scale Dilation and Erosion (cont’)
-12
31
55 5945
7
-48-60-50-40-30-20-10
010203040506070
0 1 2 3 4 5 6 7 8 9 10
F
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DC & CV Lab.DC & CV Lab.CSIE NTU
5.5.1Gray Scale Dilation and Erosion (cont’)
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DC & CV Lab.DC & CV Lab.CSIE NTU
5.5.1Gray Scale Dilation and Erosion (cont’)
19 23
9
-29
-48
-60
-50
-40
-30
-20
-10
0
10
20
30
0 1 2 3 4 5 6 7 8 9
F eorsion by K
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5.5.1Gray Scale Dilation and Erosion (cont’)
=lena.im.ero=
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DC & CV Lab.DC & CV Lab.CSIE NTU
5.5.1Gray Scale Dilation and Erosion (cont’)
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DC & CV Lab.DC & CV Lab.CSIE NTU
5.5.1Gray Scale Dilation and Erosion (cont’)
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5.5.2 Umbra Homomorphism Theorems
surface and umbra operations: inverses of each other, in a certain sense
surface operation: left inverse of umbra operation
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5.5.2 Umbra Homomorphism Theorems
Proposition 5.1
Proposition 5.2
Proposition 5.3
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5.5.3 Gray Scale Opening and Closing
gray scale opening of f by kernel k denoted by f k
gray scale closing of f by kernel k denoted by f k
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5.5.3 Gray Scale Opening and Closing (cont’)
=lena.im.open=
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5.5.3 Gray Scale Opening and Closing (cont’)
=lena.im.close=
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5.5.3 Gray Scale Opening and Closing (cont’)
duality of gray scale, dilation erosion duality of opening, closing
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5.5.3 Gray Scale Opening and Closing (cont’)
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joke
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5.6 Openings Closings and Medians
median filter: most common nonlinear noise-smoothing filter
median filter: for each pixel, the new value is the median of a window
median filter: robust to outlier pixel values leaves, edges sharp
median root images: images remain unchanged after median filter
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5.7 Bounding Second Derivatives
opening or closing a gray scale image simplifies the image complexity
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5.8 Distance Transform and Recursive Morphology
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5.8 Distance Transform and Recursive Morphology (cont’)
Fig 5.39 (b) fire burns from outside but burns only downward and right-ward
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5.9 Generalized Distance Transform
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5.10 Medial Axis
medial axis transform medial axis with distance function
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5.10.1 Medial Axis and Morphological Skeleton
morphological skeleton of a set A by kernel K ,where
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5.10.1 Medial Axis and Morphological Skeleton (cont’)
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5.10.1 Medial Axis and Morphological Skeleton (cont’)
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5.10.1 Medial Axis and Morphological Skeleton (cont’)
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5.11 Morphological Sampling Theorem
Before sets are sampled for morphological processing, they must be morphologically simplified by an opening or a closing .
Such sampled sets can be reconstructed in two ways: by either a closing or a dilation.
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=========== Joke===========
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DC & CV Lab.DC & CV Lab.CSIE NTU
5.12 Summary
morphological operations: shape extraction, noise cleaning, thickening
morphological operations: thinning, skeletonizing
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DC & CV Lab.DC & CV Lab.CSIE NTU
Homework
Write programs which do binary morphological dilation, erosion, opening, closing, and hit-and-miss transform on a binary image (Due Nov. 1)
Write programs which do gray scale morphological dilation, erosion, opening, and closing on a gray scale image (Due Nov. 15)