Digital Image Processing

1. Morphological Image Processing Digital Image Processing Bundit Thipakorn, Ph.D. Computer Engineering Department

2. Mathematical Morphology Mathematical morphology is a tool that: • is based on set theory; • concerns the study of form and structure (shape and properties of objects/regions of an image); • can be used for pre/post-processing within image analysis such as filtering, thinning, and pruning.

3. Morphology = empty set Ø = is a member or element of = ‘implies that’ Cont’d. Basic Definitions: 1. Standard set notation: The expression {p | <condition(s) > } means ‘the set of points, p, such that the listed <condition(s)> is (are) true’.

4. Morphology Cont’d. 2. ‘Included’ The binary object can be identified as a subset of pixels A contained or ‘included’ within the total set of image pixels N. That is, Therefore if p is an element of A then p is also an element of N. That is,

5. Morphology Cont’d. 3. ‘Complement’ The ‘complement’ of the set A is the set which are not element of A, that is, all the white background pixels in the image. The complement of A can be denoted as Ac and defined as: Therefore, if p is an element of A it cannot be an element of Ac, i.e.

6. Morphology Cont’d. 4. ‘Transposition (Reflection)’ The ‘transposition’ of the set of pixels, A, is the set such that each pixel position is reflected about a defined origin. The ‘transposition’ of A, denoted Â, is defined as: 5. ‘Union’ The ‘union’ of two sets A and B is denoted by A  B and defined as:

7. Morphology Cont’d. 6. ‘Intersection’ The ‘intersection’ of two sets A and B is denoted by A  B and defined as: 7. ‘Difference’ The ‘difference’ between two sets A and B is denoted by A\B and defined as:

8. Morphology Cont’d. This is analogous to the logical XOR (exclusive OR) operation. The resulting set is the set such that each pixel is belonged to either set A or set B. 8. ‘Translation’ The ‘translation’ of A by x = (x1, x2), denoted (A)x, is defined as: A vector addition

9. Morphology Cont’d. Original Image Reflection of set A to produce set B.

10. Morphology Cont’d.

11. Morphology Cont’d. A\B (A XOR B) Translation, (A)x, of set A by vector x (2,2)

12. Morphology Cont’d. • Set in mathematical morphology represent the shapes of objects in an image. • A binary image can be treated as a 2D point set of integer pairs (Z2). Each element of a set is a tuple (2-D vector) whose coordinates are the (x,y) coordinates of a black (by convention) pixel in the image. X = the origin (0,0) Y A discrete image ‘P’ = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1)}. An example of point set

13. Morphology Cont’d. • Gray-scale digital images can be represented as sets of integer triples (Z3); (x, y, intensity value). • Morphological operators usually take a binary image (point set A) and another small point set B called a “structuring element”. • B is expressed with respect to a local origin called the “representative point” and the pixel in the image A corresponding to the representative point is called the “current” pixel. • Usually, the structuring element is sized 3x3 and has its origin at the center pixel.

14. Morphology Cont’d. Generic morphological operator: • Determine the shape of structuring element. • The structuring element is shifted over the image. • At each pixel of the image elements of the structuring element are compared with the set of the underlying pixels. • If the two sets of elements match the condition defined by the set operator, the pixel underneath the origin of the structuring element is set to a pre-defined value (0 or 1 for binary image).

15. Dilation and Erosion • Two operations which are fundamental to morphological analysis of images are ‘dilation’ and ‘erosion’. • Almost all morphological operations can be defined in terms of these two basic operations. Dilation (Grow, Expand) Enlarge the boundaries of regions of foreground pixels and fill in holes in the object (a binary image).

16. Dilation Cont’d. Let A and B as sets in Z2. The dilation of A by B, denoted A  B, is defined as: Dilation i.e. An output pixel will be set at all points where the translated B (structuring element) and the image A have a ‘non-empty’ intersection. Generally, the dilation process is performed by laying the structuring element on the image and shifting it across the image in a manner similar to convolution. At each pixel,

17. Dilation Cont’d. • If the origin of the structuring element coincides with a ‘0’ in the image, there is no change; move to the next pixel. • If the origin of the structuring element coincides with a ‘1’ in the image, perform the OR logic operation on all pixels within the structuring element.

18. Dilation Cont’d. 3x3 Structuring Element (a) An Original Image (b)The Dilation of (a)

19. Erosion Erosion (Shrink, Reduce) Shrink the boundaries of regions of foreground pixels and enlarge holes in the object (a binary image). Let A and B as sets in Z2. The dilation of A by B, denoted A B, is defined as: Erosion

20. Erosion Cont’d. i.e. The erosion of A by B is simply the set of all point of A such that B is a subset of A. Generally, the erosion process is similar to dilation, but we turn pixels to ‘0’, not ‘1’. At each pixel, • If the origin of the structuring element coincides with a ‘0’ in the image, there is no change; move to the next pixel. • If the origin of the structuring element coincides with a ‘1’ in the image, and any of the ‘1’ pixels in the structuring element extend beyond the object

21. Erosion Cont’d. (‘1’ pixels) in the image, then change the ‘1’ pixel in the image to a ‘0’. Erosions can be made directional by using less symmetrical structuring elements. For example, a structuring element that is 10 pixels wide and 1 pixel high will erode in a horizontal direction only.

22. Erosion Cont’d. 3x3 Structuring Element (a) An Original Image (b)The Erosion of (a)

23. Opening and Closing The opening of an image A by the structuring element B is denoted by and is defined as Opening Erosion followed by a dilation using the same structuring element. Opening can be used to eliminate all pixels in regions that are too small to contain the structuring element.

24. Opening and Closing The closing of an image A by the structuring element B is denoted by and is defined as Cont’d. Closing Dilation followed by an erosion using the same structuring element. Closing connects objects that are close to each other, fills up small holes, and smoothes the object outline by filling up narrow gulfs.

25. Opening and Closing Cont’d. 3x3 Structuring Element (a) An Original Image (b)The Opening of (a)

26. Opening and Closing Cont’d. 3x3 Structuring Element (a) An Original Image (b)The Closing of (a)

27. Hit-or-Miss Transformation = size of the structuring element The hit-or-miss transformation of an image A by the structuring element B = (B1, B2) is denoted by and is defined as * * Hit-or-Miss Shape detection or Find local patterns of pixels. where B1 = X = pattern of pixels that we want to find, B2 = W - X, W = a small window enclosing “X”.

28. Hit-or-Miss Cont’d.

29. Boundary Extraction b(A) = A - (A B) The boundary of set A, b(A), can be obtained by the following operations:

30. Thinning and Thickening The thinning of set A by a structuring element B, denoted A B, can be defined as: A B = A - (A B) * If a sequence of structuring elements {B} = {B1, B2, B3, …, Bn} is used where Bi is a rotated version of Bi-1. The process is to thin A by one pass with B1, then thin the result with one pass of B2, and so on, unit A is thinned with one pass of Bn.

31. Thinning

32. Thickening The thickening of set A by a structuring element B, denoted A B, can be defined as: * A B = A U (A B)