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Provides mathematical tools for shape analysis in both binary and grayscale images
Chapter 13 – Mathematical Morphology
Usages: (i) Image pre-processing – noise removal, shape simplification(ii) Enhancement of object structure – skeletonizing, thinning, thickening, convex hull(iii) Object segmentation(iv) Quantitative description of objects – area, perimeter, Euler-Poincare characteristic
13.1 Basic Morphological Concepts
13-1
Morphological approach consists of 2 main steps:
(i) geometrical transformation , (ii) measurement
13.2 Morphological Principles
4 principles:
(1) Compatibility with translation -- If depends on
( ) ( ( ))h hX X the position of origin O, ;
( ) ( ( ))O h h hX X
otherwise,
(2) Compatibility with change of scale – If depends
on parameter , ;
1
( ) ( )X X
( ) ( )X X
otherwise,
13-2
(4) Upper semi-continuity – Morphological transformation does not exhibit any abrupt changes
(3) Local knowledge – only a part of a structure can
be examined,( ( )) ( )X Z Z X Z
13.3 Binary Dilation and Erosion ◎ Basic Morphological Operations
○ Duality
( )X*( ) ( ( ))c cX X *( ) :X
○ Translation2{ : , }
or { | }
X X
X
h p p = x h x
h x x
:X h
2 : 2D space13-3
Example: h = (2,2)
{ | }X X x x
○ Transposition -- Reflects a set of pixels w.r.t.
the originX
13-4
13.3.1 Dilation: an image shape, : a structuring elementX B
2{ : , and }
or { | , }
ˆ ˆ { | ( ) } { | (( ) ) }
X B B
X B
B X B X X
x x
p p x b x X b
x b x b
x x
13-5
Dilation of X by B:
can be obtained by replacing every x in X with a B
Properties:
,X B B X ,B
X B X
bb
( )h hX B X B
If then X Y X B Y B
。 It may be that
{(7,3),(6,2),(6,4),
(8,2),(8,4)}
B
X B
X X B
13-6
13.3.2 Erosion2{ : , }
or { | }B
X B B
B X X
b bb
p p x b X b
b Steps: (i) Move B over X, (ii) Find all the places where B fits (iii) Mark the origin of B when fitting
13-7
。 Erosion thins an shape
。 The origin of B may not be in B and X B X
。 Contours can be obtained by subtraction of an eroded shape from its original
13-8
○ Dilation and erosion are inverses of each other, i.e.,
。 Duality
i. The complement of an erosion equals the
dilation of the complement
where is the reflection of BB
( )C CX B X B
ii. Exchange the erosion and dilation of the above
equation ( )C CX B X B
○ Neither erosion nor dilation is an invertible transformation
13-9
。 Proof of
From the definition of erosion,
Its complement:
If , then
{ | }X B B X ww
{ | }CX B B X ww
B XwCB X w
{ | } { | }
{ | }
C C C
C C
X B B X B X
B X X B
w w
w
w w
w
( { | ( ) })X B B X ww
13-10
( )C CX B X B
◎ Boundary Detection
Let B: Symmetric about its origin The boundary of X (i) Internal boundary: -- Pixels in A that are at its edge (ii) External boundary: -- Pixels outside X that are next to it (iii) Gradient boundary: -- a combination of internal and external boundary pixels
( )X X B
( )X B X
( ) ( )X B X B
13-11
Internal boundary
external boundary
gradient boundary
13-12
Internal boundary
external boundary gradient boundary13-13
Properties: ( ) , ( ) ,h h h hX B X B X B X B
If then X Y X B Y B If then D B X B X D ( )C CX Y X Y
( ) ( ) ( )X Y B X B Y B ( ) ( ) ( )B X Y B X B Y
( ) ( ) ( ) ( )X Y B B X Y X B Y B ( ) ( ) ( ) ( )B X Y X Y B X B Y B
( ) ( ) ( )X Y B X B Y B ( ) ( ) ( )B X Y X B Y B
( ) ( )X B D X B D ( ) ( )X B D X B D
13-14
13.3.3 Hit-or-Miss Transformation-- Find shapes
: the shape to be found
: fits around
1 2
1 2
1 2 1 2
{ : , }
or ( ) ( )
( ) \ ( ), ( , )
c
c
X B B X B X
X B X B
X B X B B B B
x
1B
1B2B
。 Example – find the square in an image
13-15
1(i) X B
2(ii) cX B
1 2(iii) ( ) ( )cX B X B
13-16
○ Opening of X by B
( ) { | }X B X B B B B X w w
13.3.4 Opening and Closing
13-17
。 Properties:
(i)
(ii) Idempotence:
(iii)
X B X( )X B B X B
If , then ( ) ( )X Y X B Y B
(iv) Opening tends to (a) smooth image, (b) break narrow joins (c) remove thin protrusions
13-18
○ Closing of X by B: ( )X B X B B
。 Properties:
(i)
(ii) Idempotence:
(iii)
( )X X B
( )X B B X B
If , then ( ) ( )X Y X B Y B (iv) Closing tends to (a) smooth image, (b) fuse narrow breaks (c) thin gulfs, (d) remove small holes
13-19
13-20
( ) ,C CX B X B
( )C CX B X B
○ Properties:• Opening and closing are invariant to translation• Opening and closing are dual transformations
13.4 Gray-Scale Dilation and Erosion
[ ]( ) max{ ,( , ) }T A y y A x x
nA 。 The top-surface of set[ ]T A○ Dilation
13-21
13-22
。 The umbra of the top-surface of set A
[ ] {( , ) , ( )}U f y F y f x x
Let [ ].f T A
The umbra of function
[ ]T A
[ [ ]] {( , ) , [ ]( )}U T A y F y T A x x
1: , nf F F
The umbra of is[ ]T A
Example:
13-23
。 The dilation of f by k:
{ [ ] [ ]},f k T U f U k 1, ,nF K : , :f F k K
where
max{ ( ) ( ), , }f k f x z k z z K x z F or
。 Another illustration
For each p of X (i) Find its neighborhood according to the domain of B (ii) Compute , (iii) p = max{ }
Recall binary dilation
BDpN
pN BpN B
13-24
Example:
Final result:
13-25
( , )( )( , ) max { ( , ) ( , )}
Bs t DX B x y X x s y t B s t
。 Gray-scale Dilation:
For each pixel p of X,
(i) Lie the origin of B over p (ii) Find corresponding to B (iii) p = max{ }
pNpN B
Dilation increases light areas in an image
13-26
○ Erosion
{ [ ] [ ]},f k T U f U k
min{ ( ) ( ), , }z K
f k f x z k z z K x z F
or
13-27
(i) Move B over X,
(ii) Find all the places where B fits
(iii) Mark the origin of B when fitting
{ | }X B X B X xx
。 Another illustration
Recall binary erosion
13-28
For each p of X
(i) Find its neighborhood according to the domain of B (ii) Compute , (iii) p = min{ }pN B
BD
pN B
pN
13-29
( , )
1 , 1
( )(1,1) min { (1 ,1 ) ( , )}
min { (1 ,1 ) ( , )}Bs t D
s t
X B X s t B s t
X s t B s t
The value of X(1+s, 1+t) – B(s, t)
Minimum = 5
○ Example:
13-30
Final result:
13-31
( , )( )( , ) min { ( , ) ( , )}
Bs t DA B x y A x s y t B s t
。 Grayscale erosion:
For each pixel p of A,
(i) Lie the origin of B over p (ii) Find corresponding to B (iii) p = min{ }
pNpN B
Erosion decreases light areas in an image
13-32
( ) ( )A B A B
3 × 3 square 5 × 5 square
◎ Edge Detection
◎ Remove impulse noise
(1) removes black pixels but enlarges holes
(2) fills holes but enlarges objects
(3) reduces size
Square Cross
A B(( ) )A B B B
((( ) ) ) (( ) )A B B B B A B B 13-33
13.4.2 Umbra Homeomorphism Theorem, Properties of Erosion and Dilation, Opening and Closing
Umbra Homeomorphism Theorem:
[ ] [ ] [ ]U f k U f U k
[ ] [ ] [ ]U f k U f U k
Grey-scale opening: ( )f k f k k
Grey-scale closing: ( )f k f k k
Duality: ( )( ) (( ) )( )f k f k x x
13-34
max{ } min{ }X Y X Y Let X, Y: matrices,
1 2 3 7 6 1
4 5 6 , 8 5 2
7 8 9 9 4 3
X Y
6 4 2
max{ } max{ 4 0 4 } 6
2 4 6
X Y
6 4 2
min{ } min{ 4 0 4 } 6
2 4 6
X Y
+, -: componentwise addition and subtraction
e.g.,
13-35
max{ } , min{ } X Y A B X Y A B
13.4.3 Top Hat Transformation
-- For segmenting objects in images
\ ( )X X K
13-36
\ : subtractionwhere
max{ } min{ }X Y X Y
( ), ( )A B A B A B A B or ( ) , ( )A B A B A B A B
13.5 Skeletons and Object Marking
Homotopic transformation: a transformation doesn’t change the continuity relation between regions and holes.
13-37
13.5.1 Homotopic transformations
A transformation is homotopic if it doesn’t change the homotopic tree.
13-38
13.5.2 Skeleton, Maximal Ball
Meaning of skeleton (or medial axis):
Points where two ormore firefronts meet
Points lie on the trajectory of centers of maximal balls
Skeleton by maximal balls:
( ) { : 0, ( , ) : a maximal ball of }S X p X r B p r X
Until ( ) , Skeleton = (differences)X kB B
Erosions Openings differences
( )
( ) ( ) (( ) )
2 ( 2 ) ( 2 ) (( 2 ) )
3 ( 3 )
X X B X X B
X B X B B X B X B B
X B X B B X B X B B
X B X B B
( 3 ) (( 3 ) )
( ) ( ) (( ) )
X B X B B
X kB X kB B X kB X kB B
0
( ) ( ) \ ( )n
S X X nB X nB B
: the ball of radius nnB B B B
where
○ Lantuejoul’s method
13-39
Structuring element
Final result
13-40
Examples:
13.5.3 Thinning and ThickeningThinning: \ ( ),X B X X B Thickening: ( )X B X X B
where 1 2( , )B B B
Duality: ( )c cX B X B 13-41
Sequential thinning with 1 2( , ),L L L
1
0 0 0
1 ,
1 1 1
L
2
0 0
1 1 0
1
L
where
Sequential thinning with 1 2( , ),E E E
1
1
0 1 0 ,
0 0 0
E
2
0
0 1 0
0 0 0
E
where
13-42
13-43
13-44
13.5.4 Quench Function, Ultimate Erosion• Quench function reconstructs X as a union of its
maximal balls B.( )
( ( ) )Xp S X
X p q p B
where ( ) :S X skeleton of X
( ), ( ) :Xq p p S X ball of radius
• Global maximum, global minimum, local maximum, regional maximum
13-45
• Ultimate erosion Ult(X ): the set of regional maxima of the quench function
13-46
13.5.5 Ultimate Erosion and Distance Functions
Ult( ) ( ) \ ( ( 1) )X nBn N
X X nB X n B
where ( ) :A B the reconstruction of A from B
Ultimate erosion:
dist ( ) min{ , not in ( )}X p n N p X nB Distance function:
Influence zone:2( ) { , , ( , ) ( , )}i i jZ X p Z i j d p X d p X
13-47
Skeleton by influence zone SKIZ: the set of
boundaries of influence zones { ( )}iZ X
13.5.6 Geodesic TransformationsAdvantages:
They operate only on some part of an image
Their structuring element can vary at each pixel
( , ) :Xd x yLet geodesic distance constrained in X
13-48
The geodesic ball of center p X and radius n
( , ) { , ( , ) }X XB p n p X d p p n
The geodesic dilation of size n of Y inside X( ) ( ) ( , ) { , , ( , ) }nX X X
p Y
Y B p n p X p Y d p p n
The geodesic erosion of size n of Y inside X
( ) ( ) { , ( , ) }
{ , \ , ( , ) }
nX X
X
Y p Y B p n Y
p Y p X Y d p p n
13-49
13.5.7 Morphological Reconstruction
Reconstruction of the connected components of X
that were marked by Y.( )( ) lim ( )n
X Xn
Y Y
For binary images,
13-50
For grey scale images, considering increasing
transformations , i.e.,2, , ( ) ( )X Y Z Y X Y X
A grey-level image is viewed as a stack of binary
images obtained by successive thresholding.
( ) { , ( ) }, 0, ,k IT I p D I P k k N Thresholded grey scale image I:
13-51
Threshold decomposition principle: , ( )( ) max{ [0, , ], ( ( ))}I kp D I p k N p T I
The reconstruction of I from J
, ( )( ) max{ [0, ], ( ( ))}kI T Kp D J p k N p T J
where D : the domain of I and J
Thresholded images obey the inclusion relation
1[1, ], ( ) ( )k kk N T I T I
13-52
13.6 Granulometry
13-53
13.7 Morphological Segmentation and Watersheds
13.7.1 Particles Segmentation, Marking, and Watersheds
13.7.2 Binary Morphological Segmentation
13-54
13-55
13-56
13.7.3 Grey-Scale Segmentation, Watersheds