1. 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 13-0

2. 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 the position of origin O, ; otherwise, (2) Compatibility with change of scale – If depends on parameter, ; otherwise, 13-1

3. (4) Upper semi-continuity – Morphological transformation does not exhibit any abrupt changes (3) Local knowledge – only a part of a structure can be examined, 13.3 Binary Dilation and Erosion ◎ Basic Morphological Operations ○ Duality ○ Translation : 2D space 13-2

4. Example: h = (2,2) ○ Transposition -- Reflects a set of pixels w.r.t. the origin 13-3

5. 13.3.1 Dilation 13-4 Dilation of X by B:

6. can be obtained by replacing every x in X with a B Properties: 。 It may be that 13-5

7. 13.3.2 Erosion Steps: (i) Move B over X, (ii) Find all the places where B fits (iii) Mark the origin of B when fitting 13-6

8. 。 Erosion thins an shape 。 The origin of B may not be in B and 。 Contours can be obtained by subtraction of an eroded shape from its original 13-7

9. ○ 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 B ii. Exchange the erosion and dilation of the above equation ○ Neither erosion nor dilation is an invertible transformation 13-8

10. 。 Proof of From the definition of erosion, Its complement: If, then 13-9

11. ◎ 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 13-10

12. Internal boundary external boundary gradient boundary 13-11

13. Internal boundary external boundarygradient boundary 13-12

14. Properties: 13-13

15. 13.3.3 Hit-or-Miss Transformation -- Find shapes : the shape to be found : fits around 。 Example – find the square in an image 13-14

16. 13-15

17. ○ Opening of X by B 13.3.4 Opening and Closing 13-16

18. 。 Properties: (i) (ii) Idempotence: (iii) (iv) Opening tends to (a) smooth image, (b) break narrow joins (c) remove thin protrusions 13-17

19. ○ Closing of X by B: 。 Properties: (i) (ii) Idempotence: (iii) (iv) Closing tends to (a) smooth image, (b) fuse narrow breaks (c) thin gulfs, (d) remove small holes 13-18

20. 13-19 ○ Properties: Opening and closing are invariant to translation Opening and closing are dual transformations

21. 13.4 Gray-Scale Dilation and Erosion 。 The top-surface of set ○ Dilation 13-20

22. 13-21 。 The umbra of the top-surface of set A Let The umbra of function The umbra of is Example:

23. 13-22 。 The dilation of f by k: where or

24. 。 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 13-23

25. Example: Final result: 13-24

26. 。 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{ } Dilation increases light areas in an image 13-25

27. ○ Erosion or 13-26

28. (i) Move B over X, (ii) Find all the places where B fits (iii) Mark the origin of B when fitting 。 Another illustration Recall binary erosion 13-27

29. For each p of X (i) Find its neighborhood according to the domain of B (ii) Compute, (iii) p = min{ } 13-28

30. The value of X(1+s, 1+t) – B(s, t) Minimum = 5 ○ Example: 13-29

31. Final result: 13-30

32. 。 Grayscale erosion: For each pixel p of A, (i) Lie the origin of B over p (ii) Find corresponding to B (iii) p = min{ } Erosion decreases light areas in an image 13-31

33. 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 SquareCross 13-32

34. 13.4.2 Umbra Homeomorphism Theorem, Properties of Erosion and Dilation, Opening and Closing Umbra Homeomorphism Theorem: Grey-scale opening: Grey-scale closing: Duality: 13-33

35. Let X, Y: matrices, +, -: componentwise addition and subtraction e.g., 13-34

36. 13.4.3 Top Hat Transformation -- For segmenting objects in images 13-35 \ : subtraction where

37. 13.5 Skeletons and Object Marking Homotopic transformation: a transformation doesn’t change the continuity relation between regions and holes. 13-36 13.5.1 Homotopic transformations A transformation is homotopic if it doesn’t change the homotopic tree.

38. 13-37 13.5.2 Skeleton, Maximal Ball Meaning of skeleton (or medial axis): Points where two or more firefronts meet Points lie on the trajectory of centers of maximal balls Skeleton by maximal balls:

39. : the ball of radius n where ○ Lantuejoul’s method 13-38

40. Structuring element Final result 13-39

41. Examples: 13.5.3 Thinning and Thickening Thinning: Thickening: where Duality: 13-40

42. Sequential thinning withwhere Sequential thinning with where 13-41

43. 13-42

44. 13-43 13.5.4 Quench Function, Ultimate Erosion Quench function reconstructs X as a union of its maximal balls B. whereskeleton of X ball of radius Global maximum, global minimum, local maximum, regional maximum

45. 13-44 Ultimate erosion Ult(X ): the set of regional maxima of the quench function

46. 13-45 13.5.5 Ultimate Erosion and Distance Functions where the reconstruction of A from B Ultimate erosion: Distance function: Influence zone:

47. 13-46 Skeleton by influence zone SKIZ: the set of boundaries of influence zones 13.5.6 Geodesic Transformations Advantages: They operate only on some part of an image Their structuring element can vary at each pixel Let geodesic distance constrained in X

48. 13-47 The geodesic ball of center and radius n The geodesic dilation of size n of Y inside X The geodesic erosion of size n of Y inside X

49. 13-48 13.5.7 Morphological Reconstruction Reconstruction of the connected components of X that were marked by Y. For binary images,

50. 13-49 For grey scale images, considering increasing transformations, i.e., A grey-level image is viewed as a stack of binary images obtained by successive thresholding. Thresholded grey scale image I:

51. 13-50 Threshold decomposition principle: The reconstruction of I from J where D : the domain of I and J Thresholded images obey the inclusion relation

52. 13-51 13.6 Granulometry

53. 13-52 13.7 Morphological Segmentation and Watersheds 13.7.1 Particles Segmentation, Marking, and Watersheds 13.7.2 Binary Morphological Segmentation

54. 13-53

55. 13-54

56. 13-55 13.7.3 Grey-Scale Segmentation, Watersheds