1. Image Processing and Analysis (ImagePandA)
7 – Morphological Image Processing
Christoph Lampert / Chris Wojtan
Based on content from “Digital Image Processing” by Gonzalez and Woods
TexPoint fonts used in EMF.
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2. Outline Preliminaries Erosion and Dilation Opening and Closing
The Hit-or-Miss Transformation
Some Basic Morphological Algorithms
Gray-Scale Morphology

3. Outline Preliminaries Erosion and Dilation Opening and Closing
The Hit-or-Miss Transformation
Some Basic Morphological Algorithms
Gray-Scale Morphology

4. Preliminaries Set theory 𝐵 = 𝑤 𝑤=−𝑏, for 𝑏∈𝐵 Reflection
𝐵 𝑧 = 𝑐 𝑐=𝑏+𝑧, for 𝑏∈𝐵 Translation

5. Preliminaries Set theory Structuring elements (SEs)
Small sets or subimages used to probe an image
SEs are padded to rectangular images

6. Outline Preliminaries Erosion and Dilation Opening and Closing
The Hit-or-Miss Transformation
Some Basic Morphological Algorithms
Gray-Scale Morphology

7. Erosion and Dilation
Erosion
𝐴⊖𝐵={𝑧| 𝐵 𝑧 ⊆𝐴}
𝐴⊖𝐵={𝑧| 𝐵 𝑧 ∩ 𝐴 𝑐 =∅}

8. Erosion and Dilation
Erosion
𝐴⊖𝐵={𝑧| 𝐵 𝑧 ⊆𝐴}
𝐴⊖𝐵={𝑧| 𝐵 𝑧 ∩ 𝐴 𝑐 =∅}

9. Erosion and Dilation
Erosion
𝐴⊖𝐵={𝑧| 𝐵 𝑧 ⊆𝐴}
𝐴⊖𝐵={𝑧| 𝐵 𝑧 ∩ 𝐴 𝑐 =∅}

10. Erosion and Dilation
𝐴⊕𝐵={𝑧| 𝐵 𝑧 ∩𝐴≠∅}
Dilation

11. Erosion and Dilation
𝐴⊕𝐵={𝑧| 𝐵 𝑧 ∩𝐴≠∅}
Dilation

12. Erosion and Dilation Duality
Erosion is just like dilating the compliment
Dilation is just like eroding the compliment
(𝐴⊖𝐵) 𝑐 = 𝐴 𝑐 ⊕ 𝐵
(𝐴⊕𝐵) 𝑐 = 𝐴 𝑐 ⊖ 𝐵

13. Outline Preliminaries Erosion and Dilation Opening and Closing
The Hit-or-Miss Transformation
Some Basic Morphological Algorithms
Gray-Scale Morphology

14. Opening and Closing
Opening A∘𝐵= 𝐴⊖𝐵 ⊕𝐵
Closing A∘𝐵= 𝐴⊕𝐵 ⊖𝐵

15. Opening and Closing
Opening A∘𝐵= 𝐴⊖𝐵 ⊕𝐵
Closing A∘𝐵= 𝐴⊕𝐵 ⊖𝐵

16. Opening and Closing

17. Opening and Closing

18. Outline Preliminaries Erosion and Dilation Opening and Closing
The Hit-or-Miss Transformation
Some Basic Morphological Algorithms
Gray-Scale Morphology

19. The Hit-or-Miss Transformation
Given disjoint structuring elements 𝐶 and 𝐷
𝐴⊛𝐵= 𝐴⊖𝐶 ∩( 𝐴 𝑐 ⊖𝐷)
Point 𝑧 belongs to the hit-or-miss transform output if: 
𝐶 translated to 𝑧 fits in 𝐴,
and 𝐷 translated to 𝑧 misses 𝐴 (fits the background of 𝐴)
Result is the set of positions where the first structuring element fits in the foreground of the input image, and the second structuring element misses it completely.
It is used to detect patterns or shapes

20. The Hit-or-Miss Transformation

21. Outline Preliminaries Erosion and Dilation Opening and Closing
The Hit-or-Miss Transformation
Some Basic Morphological Algorithms
Gray-Scale Morphology

22. Boundary Extraction
𝛽 𝐴 =𝐴−(𝐴⊖𝐵)

23. Boundary Extraction
𝛽 𝐴 =𝐴−(𝐴⊖𝐵)

24. Hole Filling Pick a pixel inside of a hole, call it 𝑋 0
Iterate 𝑋 𝑘 =( 𝑋 𝑘−1 ⊕𝐵)∩ 𝐴 𝑐

25. Hole Filling Pick a pixel inside of a hole, call it 𝑋 0
Iterate 𝑋 𝑘 =( 𝑋 𝑘−1 ⊕𝐵)∩ 𝐴 𝑐

26. Extracting Connected Components
Pick a pixel inside of a shape, call it 𝑋 0
Iterate 𝑋 𝑘 =( 𝑋 𝑘−1 ⊕𝐵)∩𝐴

27. Thinning
𝐴⊗𝐵=𝐴− 𝐴⊛𝐵 =𝐴∩ (𝐴⊛𝐵) 𝑐
Repeat until no change

29. Thickening Morphological dual of thinning 𝐴⊙𝐵=𝐴∪ 𝐴⊛𝐵
Repeat until no change
More common approach:
Thin the background 𝐴 𝑐 and then compliment the result

30. Thickening

31. Pruning

32. Geodesic Dilation

33. Geodesic Erosion

34. Morphological Reconstruction

35. Morphological Reconstruction

36. Border clearing Find all boundary pixels in A
Morphologically reconstruct all of them
Subtract those from the original

37. Border clearing

38. Outline Preliminaries Erosion and Dilation Opening and Closing
The Hit-or-Miss Transformation
Some Basic Morphological Algorithms
Gray-Scale Morphology

39. Gray-Scale Morphology

40. Gray-Scale Morphology
Erosion
𝑓⊖𝑏 𝑥,𝑦 = min 𝑠,𝑡 ∈𝑏 𝑓(𝑥+𝑠,𝑦+𝑡)
Dilation
𝑓⨁𝑏 𝑥,𝑦 = max 𝑠,𝑡 ∈𝑏 𝑓(𝑥−𝑠,𝑦−𝑡)

41. Gray-Scale Morphology

42. Gray-Scale Morphology
Erosion
𝑓⊖𝑏 𝑥,𝑦 = min 𝑠,𝑡 ∈𝑏 𝑓(𝑥+𝑠,𝑦+𝑡)
Dilation
𝑓⨁𝑏 𝑥,𝑦 = max 𝑠,𝑡 ∈𝑏 𝑓(𝑥−𝑠,𝑦−𝑡)
Opening
f∘𝑏= 𝑓⊖𝑏 ⊕𝑏
Closing
f∘𝑏= 𝑓⊕𝑏 ⊖𝑏

43. Gray-Scale Morphology

44. Gray-Scale Morphology

45. Gray-Scale Morphology

46. Gray-Scale Morphology
Mostly analogous to binary morphology
Lots of potential, but lots of heuristics/hacks
Not easy to find general tricks that work on all problems

47. Gray-Scale Morphology