Dilation and Erosion

1. Morphology • Morphology is a branch of biology , deals with form and structure of animals and plants • Mathematical morphology is a tool for extracting image components useful in: • representation and description of region shape (e.g. boundaries) • pre- or post-processing (filtering, thinning, etc.) • Based on set theory

2. Morphology • Sets represent objects in images • Sets in binary images  (x,y) • Sets in gray scale images  (x,y,g) • Some morphological operations: Dilation & Erosion Opening & Closing Hit-or-Miss Transform Basic Algorithms

3. Morphological Image Processing

4. Morphological Image Processing

5. Morphological Image Processing

7. Dilation & Erosion • Basic operations • Are dual to each other - Erosion shrinks foreground and enlarges background - Dilation enlarges foreground and shrinks background

8. Dilation & Erosion • Dilation: “grows’ or “thickens” objects in an image. One of the simplest application of dilation is bridging gaps. • Erosion: Shrinking or thinning operation.

10. Example for Erosion • Input image • Structuring element • Output image

11. Example for Erosion • Input image • Structuring element • Output image

12. Example for Erosion • Input image • Structuring element • Output image

13. Example for Erosion • Input image • Structuring element • Output image

14. Example for Erosion • Input image • Structuring element • Output image

15. Example for Erosion • Input image • Structuring element • Output image

16. Example for Erosion • Input image • Structuring element • Output image

17. Example for Erosion • Input image • Structuring element • Output image

18. Erosion • Erosion is the set of all points in the image , where the structuring element “ fits into” • Consider each foreground pixel in the input image - if the structuring element fits in, write a “1” at the origin of the structuring element

21. Example for Dilation • Input image • Structuring element • Output image

22. Example for Dilation • Input image • Structuring element • Output image

23. Example for Dilation • Input image • Structuring element • Output image

24. Example for Dilation • Input image • Structuring element • Output image

25. Example for Dilation • Input image • Structuring element • Output image

26. Example for Dilation • Input image • Structuring element • Output image

27. Example for Dilation • Input image • Structuring element • Output image

28. Example for Dilation • Input image • Structuring element • Output image

29. Dilation • Dilation is the set of all points in the image , where the structuring element “touches” the foreground • Consider each pixel in the input image - if the structuring element touches the foreground image, write a “1” at the origin of the structuring element

30. Dilation & Erosion • Basic definitions: • A,B: sets in Z2 with components a=(a1,a2) and b=(b1,b2) • Translation of A by x=(x1,x2), denoted by (A)x is defined as: (A)x = {c| c=a+x, for a∈A}

31. Dilation & Erosion • More definitions: Reflection of B: = {x|x=-b, for b∈B} Complement of A: Ac = {x|xA} Difference of A & B: A-B = {x|x∈A, x  B} = A∩Bc

32. Dilation & Erosion • Dilation: • : empty set; A,B: sets in Z2 • Dilation of A by B:

33. Dilation & Erosion • Dilation: • Obtaining the reflection of B about its origin and then shifting this reflection by x • The dilation of A by B then is the set of all x displacements such that and A overlap by at least one nonzero element…

34. Dilation & Erosion • Dilation: B is the structuring element in dilation.

35. Morphological Image Processing

36. Morphological Image Processing

37. Dilation & Erosion • Erosion: i.e. the erosion of A by B is the set of all points x such that B, translated by x, is contained in A. In general:

38. Morphological Image Processing

39. Morphological Image Processing