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
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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.
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Provides mathematical tools for shape analysis in both binary and grayscale images
(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
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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