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Chapter 10: Mathematical Morphology

Chapter 10: Mathematical Morphology. Provides mathematical tools for shape analysis in both binary and grayscale images . They are suitable to be implemented by hardware. ◎ Basic Operations ○ Reflection -- Reflects a set of pixels w.r.t. the origin. 。 Example:. ○ Translation

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Chapter 10: Mathematical Morphology

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  1. Chapter 10: Mathematical Morphology • Provides mathematical tools for shape analysis in • both binaryandgrayscale images. • They are suitable to be implemented by hardware. • ◎ Basic Operations • ○Reflection • -- Reflects a set of pixels w.r.t. the origin 。 Example:

  2. ○ Translation • w: a displacement (a vector) 。 Example: w = (2,2) ○ Dilation : dilation of A by B B : a structuring element

  3. 。Example: can be obtained by replacing every x in A with a B

  4. 。Example

  5. 。 。Dilation has the effect of increasing the size of an shape 。 The origin of B may not be in B and it may be that

  6. ○ Erosion • Steps: (i) Move B over A, • (ii) Find all the placeswhere B fits • (iii) Mark the origin of B when fitting 。 Example:

  7. 。The origin of B • may not be in B • and 。Erosion thins an shape

  8. Proof: From the definition of erosion, Its complement: If , then • ○ 。 Duality (Assigment)

  9. ◎ Boundary Detection • Let B: symmetric about its origin • The boundary of A • (i) Internal boundary: • -- pixels in A that are at its edge • (ii) External boundary: • -- pixels outside A that are next to it • (iii) Gradient boundary: • -- a combination of internal and external • boundary pixels

  10. 。 Example: external boundary Internal boundary gradient boundary

  11. 。 Real image Internal boundary external boundary gradient boundary

  12. 。 Example: • ○ Opening 1) 2)

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

  14. 。 Example: • ○ Closing

  15. 。 Properties: • (i) • (ii) Idempotence: • (iii) • (iv) Closing tends to • (a) smooth image, (b) fuse narrow breaks • (c) thin gulfs, (d) remove small holes

  16. Show ○ Relationship between opening and closing Proof: (Assign.)

  17. Square • ◎ Removal of impulse (salt and pepper) noise Cross B=Cross B=Square • removes single black and white pixels • but enlarges holes (2) fills holes by dilating twice but enlarge the objects • reduces • the size by an erosion

  18. ○ Hit-or-Miss Transform -- Finding shapes : the shape to be found : fits around 。Example – find the square in an image A B = A: (i) (hit)

  19. (ii) (miss) A

  20. ◎ Region filling • A: 8-connected boundary, p : a point within A

  21. 。Example: 。Real image

  22. ◎ Connected components • Let A: a collection of components • C: a component of A • p : a point in C • B : a structuring element • Using cross-shaped B to find 4-connected • components • square B to find 8-connected • components

  23. 。Example: 。Real image 3 × 3 Structuring element 11 ×11 Structuring element

  24. ◎ Skeletonization (thinning) Applications: OCR , fingerprint recognition, map digitization.

  25. Optical Character Recognition (OCR) Fingerprint Recognition

  26. ○ Lantuejoul’s method

  27. Structuring element Final result

  28. 。 Real image • Square • structuring • element Cross structuring element

  29. ◎ Grayscale Morphology • ○Binary erosion: • (i) Move B over A, • (ii) Find all the placeswhere B fits • (iii) Mark the origin of B when fitting

  30. For each p of A (i) Find its neighborhood according to the domain of B (ii) p = min{ }

  31. 。 Example: • The value of A(1+s, 1+t) – B(s, t) Minimum = 5

  32. Final result:

  33. 。 Summary for the process of grayscale erosion: For each pixel p of A, (i) Lie the origin of B over p (ii) Find according to domain of B (iii) p = min{ }

  34. 。 Example: 5 × 5 square structuring element ＊ Erosion decreases light areas in an image

  35. ○ Binary dilation For each p of A (i) Find its neighborhood according to the domain of B (ii) p = max{ }

  36. 。 Example:

  37. Final result:

  38. 。 Summary for the process of grayscale dilation: For each pixel p of A, (i) Lie the origin of B over p (ii) Find according to domain of B (iii) p = max{ }

  39. 。 Example: 5 × 5 square structuring element ＊Dilation increases light areas in an image

  40. ◎ Relationship between grayscale erosion and dilation Let X, Y: matrices, e.g.,

  41. ○ Edge Detection 3 × 3 square 5 × 5 square

  42. ◎ Opening = erosion + dilation Closing = dilation + erosion 。 Example: 5 × 5 square structuring element Opening Closing

  43. ○ Noise Removal

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