Morphological image processing motes.pdf

notes

1. Chapter 9: Morphological
Image Processing
Digital Image Processing

2. 2
Mathematic Morphology
! used to extract image components that are
useful in the representation and description of
region shape, such as
! boundaries extraction
! skeletons
! convex hull
! morphological filtering
! thinning
! pruning

3. 3
Basic Set Theory

4. 4
Reflection and Translation
} ,|{
ˆ
Bfor bbwwB ∈−∈=
} ,|{)( Afor azaccA
z ∈+∈=

5. 5
Example

6. Structuring element (SE)
6
"  small set to probe the image under study
"  for each SE, define origo
"  shape and size must be adapted to geometric
properties for the objects

7. Basic morphological operations
! Erosion
! Dilation

! combine to
! Opening object
! Closening background
7
keep general shape but
smooth with respect to

8. Erosion
! Does the structuring element fit the
set?
erosion of a set A by structuring element
B: all z in A such that B is in A when
origin of B=z
shrink the object
8
}{ Az|(B)BA
z⊆=−

9. Erosion
9

10. Erosion
10

11. 11
Erosion
}{ Az|(B)BA
z⊆=−

12. Dilation
! Does the structuring element hit the
set?
! dilation of a set A by structuring element
B: all z in A such that B hits A when
origin of B=z
! grow the object
12
}
ˆ
{ ΦA)Bz|(BA
z
≠∩=⊕

13. Dilation
13

14. Dilation
14

15. 15
Dilation
}
ˆ
{ ΦA)Bz|(BA
z
≠∩=⊕
B = structuring element

16. 16
Dilation : Bridging gaps

17. useful
! erosion
! removal of structures of certain shape and
size, given by SE
! Dilation
! filling of holes of certain shape and size,
given by SE
17

18. Combining erosion and
dilation
! WANTED:
! remove structures / fill holes
! without affecting remaining parts
! SOLUTION:
! combine erosion and dilation
! (using same SE)
18

19. 19
Erosion : eliminating irrelevant
detail
structuring element B = 13x13 pixels of gray level 1

20. Opening
erosion followed by dilation, denoted C,
! eliminates protrusions
! breaks necks
! smoothes contour
20
BBABA ⊕−= )(

21. Opening
21

22. Opening
22

23. 23
Opening
BBABA ⊕−= )(
})(|){( ABBBA
zz ⊆∪=

24. Closing
dilation followed by erosion, denoted •


! smooth contour
! fuse narrow breaks and long thin gulfs
! eliminate small holes
! fill gaps in the contour
24
BBABA −⊕=• )(

25. Closing
25

26. Closing
26

27. 27
Closing
BBABA −⊕=• )(

28. 28
Properties
Opening
(i)  A°B is a subset (subimage) of A
(ii)  If C is a subset of D, then C °B is a subset of D °B
(iii)  (A °B) °B = A °B
Closing
(i)  A is a subset (subimage) of A•B
(ii)  If C is a subset of D, then C •B is a subset of D •B
(iii)  (A •B) •B = A •B
Note: repeated openings/closings has no effect!

29. Duality
! Opening and closing are dual with respect
to complementation and reflection
29
)
ˆ
()( BABA
cc
=•

30. 30

31. 31

32. Useful: open & close
32

33. Application: filtering
33

34. Hit-or-Miss Transformation
C? (HMT)
! find location of one shape among a set of shapes
ztemplate matching
! composite SE: object part (B1) and background
part (B2)
! does B1 fits the object while, simultaneously,
B2 misses the object, i.e., fits the background?
34

35. 35
Hit-or-Miss Transformation
)]([)( XWAXABA
c
−−∩−=∗

36. 36
Boundary Extraction
)()( BAAA −−=β

37. 37
Example

38. 38
Region Filling
,...3,2,1 )(
1
=∩⊕=
−
kABXX
c
kk

39. 39
Example