1. Morphological Image Processing

2. Morphology Morphology Mathematical Morphology
The branch of biology that deals with the form and structure of organisms without consideration of function
Mathematical Morphology
Mathematical tool for processing shapes in image, including boundaries, skeletons, convex hulls, etc.
Use of set theoretical approach
(c) by Yu Hen Hu

3. Set Theory: Definitions and Notations
A collection of objects (elements)
membership ()
If  is an element (member) of a set , we write   
Subset ()
Let A, B are two sets. If for every a  A, we also have a  B, then the set A is a subset of B, that is, A  B
If A  B and B  A, then A = B.
Empty set ()
Complement set
If A  , then its complement set Ac = {|   , and A}
Union ()
A B = {|  A or  B}
Intersection ()
A  B = {|  A and  B}
Set difference (-)
B\A = B Ac
Note that B-A A-B
Disjoint sets
A and B are disjoint (mutually exclusive) if A  B= 
(c) by Yu Hen Hu

4. Set Relations
(c) by Yu Hen Hu

5. Translation and Reflection
Translation (A)z = { c| c = a + z, for a A }
Reflection:
(c) by Yu Hen Hu

6. Logic Operations Between Binary Images
(c) by Yu Hen Hu

7. Dilation and Erosion Dilation B: structure element Erosion
A  B = {z | (B)z  A}
Relations
(A  B)c =
(c) by Yu Hen Hu

8. Example of Dilation
(c) by Yu Hen Hu

9. Example of Erosion
(c) by Yu Hen Hu

10. Opening
A  B = (A  B)  B
(c) by Yu Hen Hu

11. Closing
A  B = (A  B)  B
(c) by Yu Hen Hu

12. Example: Opening & Closing
(c) by Yu Hen Hu

13. Finger Print Processing using Opening and Closing
(c) by Yu Hen Hu

14. Hit-or-Miss Transformation for shape detection
Figure 9.12 (a) Set A, (b) A window W and the local
Background of X w.r.t. W, W-X. (c) Ac. (d) AX
Intersection of (d) and (e) shows the location
of the origin of X, as desired.
(c) by Yu Hen Hu

15. Hit-or-Miss Transform
Denote B1: object, B2: local background of B1, then, or
Reason to have a local background:
Two or more objects are distinct only if they form disjoint (disconnected) sets. This is guaranteed by requiring that each object have at least a one-pixel-thick background around it.
(c) by Yu Hen Hu

16. Hit-or-Miss Transform
Previous example does not contain don’t care entries.
In structure element
1 – foreground
0 – background
X – don’t care
Output is 1 if exact match of both foreground and background pixels.
Hitnmiss.m
+1: foreground
-1: background
0: don’t care
Hitnmiss.m
(c) by Yu Hen Hu

17. Morphological Boundary Extraction
(A) = A − (A  B) (9.5-1)
(c) by Yu Hen Hu

18. Example of Boundary Extraction
(c) by Yu Hen Hu

19. Region Filling
Fig915.m
(c) by Yu Hen Hu

20. Region Filling Example
(c) by Yu Hen Hu

21. Connected Component Extraction
Y: connected component in set A,
p: a known point in Y
Fig915.m
(c) by Yu Hen Hu

22. Thinning
Thinning is often accomplished using a sequence of rotated structuring elements (a). Given a set A (b), results of thinning with first element is shown in (c), and the next 7 elements (d) – (i). There is no change between 7th and 8th elements, and no change after first 3 elements. Then it converges to a m-connectivity.
Fig921.m
(c) by Yu Hen Hu

23. Thickening AB = A hitnmiss(A,B) A{B} =((…(AB1) B2) … Bn)
Thickening is the dual of thinning
operation. Usually, thickening a set A is accomplished by thinning Ac, and then complement the result. Then a post-processing prunning process is applied to remove disconnected points as shown to the left.
(c) by Yu Hen Hu

24. Skeleton
A skeleton of a set A consists of points z that is the center of a maximum disk
A maximum disk is a circle in A that can not be enclosed by another circle that is also in A.
Figure (a) set A, (b), (c) sets of possible maximum disks. (d) dotted line is the skeleton.
(c) by Yu Hen Hu

25. Skeleton Equations Define k consecutive erosions of A as:
AkB = ( …(AB)B) …)B) (9.5-13)
Sk(A) = (AkB) − (AkB)B (9.5-12)
Let K = max{k | (AkB) } (9.5-14)
Then the skeleton can be found as:
(c) by Yu Hen Hu

26. Illustration of Skeleton Computation
Figure 9.24 Implementation of eq. (9.5-11)-(9.5-15). The original set is at the top left and its morphological skeleton is at the bottom of the 4th column. The reconstructed set is at the bottom of the 6th column.
Define k consecutive erosions of A as:
AkB = ( …(AB)B) …)B) (9.5-13)
Sk(A) = (AkB) − (AkB)B (9.5-12)
Let K = max{k | (AkB) } (9.5-14)
Then the skeleton can be found as:
(c) by Yu Hen Hu

27. Pruning
(c) by Yu Hen Hu