1. Binary Image Processing
Introduction
Set theory review
Morphological filtering
Erosion and dilation
Opening and closing
Hit-or-miss, boundary extraction, …
Skeleton via distance transform

2. Binary Images
Images only consist of two colors (tones): white or black
Numerical example (image of a square block)

3. Binary Image Examples

4. Why are binary images special?
Since pixels are either white or black, the locations of white(black) pixels carry ALL information of binary images
Example
L={(3,3),(3,4),(4,3),(4,4)}
location of white pixels
f(m,n)
matrix representation
set representation
It is often more convenient to consider the set representation than the matrix representation for binary images

5. Binary Image Processing
Introduction
Set theory review
Morphological filtering
Erosion and dilation
Opening and closing
Hit-or-miss, boundary extraction, …
Skeleton via distance transform

6. Set Theory Review
Think of sets A and B as the collections of spatial coordinates

7. Translation Operator
Example

8. Reflection Operator
Example ^ B B

9. Binary Image Processing
Introduction
Set theory review
Morphological filtering
Erosion and dilation
Opening and closing
Hit-or-miss, boundary extraction, …
Skeleton via distance transform

10. Structuring Element B
Definition: a set of local neighborhood with specified origin
Examples
origin B1 B2
Note: different structuring element leads to different filtering result

11. Dilation
Definition or
Example X Y
mask B

12. Illustration by AnimationX
origin B
B={a,b,c}
a=(0,0),b=(0,-1),c=(-1,0)
Xca
Xcb
Xcc

13. Erosion
Definition
Y=X B _
Example X Y
mask B

14. Illustration By Animation
mask B Y X

15. Theorem* _ ^ (X B)c =XcB _ (X B)c ^ Proof: ={z | Bz A }c
= {z | Bz Ac = }c
= {z | Bz  Ac  }
=Xc B ^

16. Example _ _ Xc
X B
(X B)c X B ^ B ^
XcB

17. Opening Operator _
Definition
(X B) B +
Example X _
mask B +

18. Geometric Interpretation of Opening Operator

19. Closing Operator _
Definition
(X B) B +
Example X +
mask B _

20. Geometric Interpretation of Closing Operator

21. Properties of Opening and Closing Operators*● ● ●
Closing ● ● ●

22. A Little Game of Matching
Templates A B

23. Illustration by a Simpler Case
How to find the match of A in X
using a computer?
Hit: the southwest quadrant must be black in X
Hit: the other three quadrants must be white in X X
Hit: the southwest quadrant must be black in X
origin
Hit: the other three quadrants must be black in Xc
Template B

24. Matching via Hit-or-Miss
Hit: the southwest quadrant must be black in X
origin _
Template B
X1=X B1
Hit: the other three quadrants must be black in Xc
Template B1 _
X2 =Xc B2
To satisfy both conditions, we need to take the
Intersection of X1 and X2
Template B2

25. Hit-or-Miss Operator * _ _ _ X B= (X B1)(Xc B2) Hit Miss origin
Why ?
Definition *
(X B1)(Xc B2)
Why complement?
Why intersection?
Hit
Miss
Structuring element example
origin
1 -1 x x x
mask B1
mask B2
mask B
MATLAB
(MATLAB function: bwhitmiss)

26. Example 1 _ _ X
X B1
Xc B2 Xc
origin
X B *
mask B1
mask B2

27. * -1 -1 -1 1 -1 1 1 -1 Example 2 mask B1 mask B2 mask B MATLAB _ _ X
origin
1 -1
Example 2
mask B1
mask B2
mask B
MATLAB _ _ X
X B1
Xc B2 Xc
Now, take the intersection
and note that only two points
Remain (highlighted by red)
X B *

28. Roadmap of Morphological Filtering
basic set operators (complement, union, intersection, difference)
dilation/erosion operators
closing/opening, Hit-or-Miss operators
morphological filters/algorithms
(boundary, region filling, thinning)

29. 1. Boundary Extraction _ X=X-(X B) X=(X B) – B? + mask B _ X X B _
Definition
X=X-(X B)
How about
Example
X=(X B) – B? +
mask B _ X
X B _
X-(X B)
MATLAB function: bwmorph(…,’remove’)

30. Image Example X X

31. 2. Region Filling P X
Idea: recursively expand the region around P but stop
the expansion at the boundary of X
Iterations:
expansion
stop at the boundary
Y0=P
mask B
Why dilation?
Why intersection with Xc?
Yk=(Yk-1B)Xc, k=1,2,3…
Terminate when Yk=Yk-1,output YkX

32. Image Example P X Y0
Y0B Xc Y1
Y1B
Y2B Y2
Y3=Y2

33. Additional Example P

34. 3. Thinning
thinning
Intuitively, thinning finds the skeleton of a binary image (you will learn a
different way of finding skeleton by distance transform later)

35. Thinning Algorithm * X0=X Xk=(…( (Xk-1  B1)  B2 …  B8) Basic idea:
 Use Hit-or-Miss operator as a sifter
 Use multiple masks to characterize different patterns x x x x x x x x x x x x x x x x B1 B2 B3 B4 B5 B6 B7 B8
X0=X
Why eight different B’s?
Why hit-or-miss?
Xk=(…( (Xk-1  B1)  B2 …  B8)
where
X  B=X – X B *
Stop the iteration when Xk=Xk-1

36. Image Example

37. Summary of Morphological Filtering
MATLAB codes
circshift(A,z)
fliplr(flipud(B))
~A or 1-A
A &~B
imdilate(A,B)
imerode(A,B)
imopen(A,B)
imclose(A,B)

38. Summary (Con’d) bwhitmiss(A,B) A&~(imerode(A,B)) region_fill.m
bwmorph(A,’thin’);

39. Binary Image Processing
Introduction
Set theory review
Morphological filtering
Erosion and dilation
Opening and closing
Hit-or-miss, boundary extraction, …
Skeleton via morphological filtering and distance transform

40. Medial Axis (Skeleton)
• Definition
Suppose that a fire line propagates with constant speed from the
contour of a connected object towards its inside, then all those
points lying in positions where at least two wave fronts of the fire
line meet during the propagation will constitute a form of a skeleton
• Examples

41. Morphological Filtering: Skeletons_ _ _ _
X kB=(((X B) B) … B)
Define
k-fold _ _
and
Sk(A)=(A kB)-(A KB) B

42. Morphological Filtering: Skeletons
Then the skeleton of an image A is given by

43. Image Example

44. Skeleton Finding via Distance Transform
• Distance transform: find the distance from the nearest boundary for each point
- initialization
- iteration

45. Skeleton Finding via Distance Transform
• Skeleton is the set of points whose distance from the nearest boundary is locally maximum

46. Numerical Example
2 2 3
local
maximum
x0(m,n)
x1(m,n)
x3(m,n)
skeleton

47. Image Example
original
skeleton
MATLAB code: y=bwmorph(x,’skel’,inf);

48. Image Example (Con’t)
Binary fingerprint image
Skeleton image