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  1. Introduction to Computer Vision Lecture 15 Morphological Processing Dr. Roger S. Gaborski

  2. In class exerciseTuesday before Thursday’s Quiz

  3. What is the equation for the covariance matrix? Cov(x,y) =

  4. What is the equation for the covariance matrix? variance (x) = covariance (x, y) = First calculate means for red, green and blue data Then calculate individual covariance values Roger S. Gaborski Roger S. Gaborski 5

  5. Covariance MATLAB: cov Diagonal: variances Matrix is symmetric Roger S. Gaborski Roger S. Gaborski 6

  6. Covariance MATLAB: cov Diagonal: variances Matrix is symmetric Roger S. Gaborski Roger S. Gaborski 7

  7. Image1 has the follow mean values: Red mean value: Mr = 5, Green mean value: Mg = 7 Blue mean value: Mb = 6. In vector format: M = [5; 7; 6] A new pixel P has the following values: R= 7, G= 6, B= 1, In vector format: P= [7;6;1]

  8. What is the equation for the Euclidean distance using the terms given above? dEuc(P,M) = = [ (P-M)T (P-M) ]1/2 or = [(Pr-MR)2 + (Pg-Mg)2 + (Pb-Mb)2]1/2 dEuc(P,M) =( [7-5,6-7,1-6]*[7-5; 6-7; 1-6] )1/2 = (30)1/2 = 5.5

  9. What is the for the Mahalanobisdistance using the values given above? Need inverse of covariance matrix Then: DM(P,M) = [ (P-M)T C-1(P-M) ]1/2

  10. DM(P,M) = [ (P-M)T C-1(P-M) ]1/2 Assume C-1 = [1 0 .5; 0 1 0; .5 0 1] First part: [7-5,6-7,1-6]*[1 0 .5; 0 1 0; .5 0 1] = [2, -1, -5] * [1 0 .5; 0 1 0; .5 0 1] = = [-.5 -1 -4] Second part: [-.5 -1 -4] *[2; -1; -5] = 20, DM(P,M) = sqrt(20)

  11. Agenda • Binary morphological processing • Erosion and dilation • Opening and closing • Gray-scale morphological processing • Erosion and dilation • Morphological gradients • Example of Background Removal Roger S. Gaborski 12

  12. Introduction • Morphology: a branch of biology dealing with the form and structure of creatures • Mathematical morphology: • Extract image components based on shape e.g. boundaries, skeletons, convex hull, etc • Image denoise e.g. reduce noise after edge detection • Remove Background variation Roger S. Gaborski 13

  13. Binary Morphological Processing Non-linear image processing technique Order of sequence of operations is important Linear: (3+2)*3 = (5)*3=15 3*3+2*3=9+6=15 Non-linear: (3+2)2 = (5)2 =25 [sum, then square] (3)2 + (2)2 =9+4=13 [square, then sum] Based on geometric structure Used for edge detection, noise removal and feature extraction Used to ‘understand’ the shape/form of a binary image Roger S. Gaborski 14

  14. Image – Set of Pixels Basic idea is to treat an object within an image as a set of pixels (or coordinates of pixels) In binary images Background pixels are set to 0 and appear black Foreground pixels (objects) are 1 and appear white Roger S. Gaborski 15

  15. Chapter 9 Morphological Image Processing A-B = A- (A∩B) From: Digital Image Processing, Gonzalez,Woods And Eddins Roger S. Gaborski 16

  16. DILATION A A1= A B B B a a Object B is one point located at (a,0) A1: Object A is translated by object B Since dilation is the union of all the translations, A B = U Atwhere the set union U is for all the b’s in B, the dilation of rectangle A in the positive x direction by a results in rectangle A1 (same size as A, just translated to the right) Roger S. Gaborski 17

  17. DILATION – B has 2 Elements A A2 A1 (part of A1 is under A2) -a a -a a Object B is 2 points, (a,0), (-a,0) There are two translations of A as result of two elements in B Dilation is defined as the UNION of the objects A1 and A2. NOT THE INTERSECTION Roger S. Gaborski 18

  18. DILATION Countless translation Vectors Rounded corners Image (A) SE (B) Dilation Roger S. Gaborski 19

  19. DILATION Another approach Image (A) SE (B) Round Structuring Element (SE) can be interpreted as rolling the SE around the contour of the object. New object has rounded corners and is larger by ½ width of the SE Dilation Roger S. Gaborski 20

  20. DILATION Countless translation vectors Square corners Image (A) SE (B) Dilation Roger S. Gaborski 21

  21. DILATION Another approach Square corners Image (A) SE (B) Square Structuring Element (SE) can be interpreted as moving the SE around the contour of the object. New object has square corners and is larger by ½ width of the SE Dilation Roger S. Gaborski 22

  22. DILATION The shape of B determines the final shape of the dilated object. B acts as a geometric filter that changes the geometric structure of A Roger S. Gaborski 23

  23. Chapter 9 Morphological Image Processing From: Digital Image Processing, Gonzalez,Woods And Eddins Roger S. Gaborski 24

  24. Chapter 9 Morphological Image Processing Image A Image B ~ A A U B A ∩B A-B = A ∩(~B) From: Digital Image Processing, Gonzalez,Woods And Eddins Roger S. Gaborski 25

  25. SE Original Image Translation Process Dilated Image From: Digital Image Processing, Gonzalez,Woods And Eddins Roger S. Gaborski 26

  26. imdilate • IM2 = IMDILATE(IM,NHOOD) dilates the image IM, where NHOOD is a • matrix of 0s and 1s that specifies the structuring element • neighborhood. This is equivalent to the syntax IIMDILATE(IM, • STREL(NHOOD)). IMDILATE determines the center element of the • neighborhood by FLOOR((SIZE(NHOOD) + 1)/2). • >> se = imrotate(eye(3),90) • se = • 0 0 1 • 0 1 0 • 1 0 0 • >> ctr=floor(size(se)+1)/2 • ctr = • 2 2 1 2 3 1 2 3 Roger S. Gaborski 27

  27. MATLAB Dilation Example Im (original image) Im2 (dialated image) >> Im = zeros([13 19]); >> Im(6,6:8)=1; >> Im2 = imdilate(Im,se); 1 2 3 Roger S. Gaborski 28 1 2 3

  28. MATLAB Dilation Example INPUT IMAGE DILATED IMAGE >> I = zeros([13 19]); >> I(6, 6:12)=1; >> SE = imrotate(eye(5),90); >> I2=imdilate(I,SE); >> figure, imagesc(I) >> figure, imagesc(SE) >> figure, imagesc(I2) 1 2 3 4 5 SE 1 2 3 4 5 Roger S. Gaborski 29

  29. MATLAB Dilation Example INPUT IMAGE DILATED IMAGE I I2 1 2 3 4 5 >> I(6:9,6:13)=1; >> figure, imagesc(I) >> I2=imdilate(I,SE); >> figure, imagesc(I2) SE 1 2 3 4 5 Roger S. Gaborski 30

  30. MATLAB Dilation Example DILATED IMAGE INPUT IMAGE I I2 SE = 1 1 1 1 1 1 1 1 1 Roger S. Gaborski 31

  31. Dilation and Erosion DILATION: Adds pixels to the boundary of an object EROSIN: Removes pixels from the boundary of an object Number of pixels added or removed depends on size and shape of structuring element Roger S. Gaborski 32

  32. EROSIN SE Original Image Translation Process Eroded Image From: Digital Image Processing, Gonzalez,Woods And Eddins Roger S. Gaborski 33

  33. MATLAB Erosion Example ERODED IMAGE 2 pixel wide INPUT IMAGE >> I=zeros(13, 19); I(6:9,6:13)=1; >> figure, imagesc(I) >> I2=imerode(I,SE); >> figure, imagesc(I2) 1 2 3 1 2 3 34 Roger S. Gaborski SE = 3x1

  34. Chapter 9 Morphological Image Processing Original Image Erosion with a disk of radius 10 From: Digital Image Processing, Gonzalez,Woods And Eddins Erosion with a disk of radius 5 Erosion with a disk of radius 20 35 Roger S. Gaborski

  35. Combinations In most morphological applications dilation and erosion are used in combination May use same or different structuring elements Roger S. Gaborski 36

  36. Morphological Opening and Closing Opening (o) of A by B A o B = (AO B)  B; imopen(A, B) Erosion of A by B, followed by the dilation of the result by B Closing ()of A by B  A B = (A  B) O B; imclose(A, B) Dilation of A by B, followed by the erosion of the result by B Roger S. Gaborski 37

  37. MATLAB Function strel strel constructs structuring elements with various shapes and sizes Syntax: se = strel(shape, parameters) Example: se = strel(‘octagon’, R); R is the dimension – see help function Roger S. Gaborski 38

  38. Opening of A by B  A B • Erosion of A by B, followed by the dilation of the result by B Erosion- if any element of structuring element overlaps with background output is 0 f (original image) fe (eroded image) FIRST - EROSION >> se = strel('square', 20); fe = imerode(f,se); figure, imagesc(fe),title('fe') Roger S. Gaborski 39

  39. Dilation of Previous Result Outputs 1 at center of SE when at least one element of SE overlaps object fe (eroded image) fd (dilated image) SECOND - DILATION >> se = strel('square', 20); fd = imdilate(fe,se); figure, imagesc(fd),title('fd') Roger S. Gaborski 40

  40. FO = imopen(f,se); figure, imagesc(FO),title('FO') FO (opened image) Original Image Roger S. Gaborski 41

  41. What if we increased size of SE for DILATION operation?? se = 25 se = 30 se = strel('square', 30); fd = imdilate(fe,se); figure, imagesc(fd),title('fd') se = strel('square', 25); fd = imdilate(fe,se); figure, imagesc(fd),title('fd') Roger S. Gaborski 42

  42. Closing of A by B  A B Dilation of A by B Outputs 1 at center of SE when at least one element of SE overlaps object se = strel('square', 20); fd = imdilate(f,se); figure, imagesc(fd),title('fd') Roger S. Gaborski 43

  43. Erosion of the result by B Erosion- if any element of structuring element overlaps with background output is 0 Roger S. Gaborski 44

  44. ORIGINAL OPENING CLOSING Roger S. Gaborski 45

  45. Chapter 9 Morphological Image Processing original image opening opening + closing From: Digital Image Processing, Gonzalez,Woods And Eddins Roger S. Gaborski 46

  46. Hit or Miss Transformation Usage: to identify specified configuration of pixels, e.g. isolated foreground pixels pixels at end of lines (end points) Definition A B = (A B1) ∩(Ac ΘB2) A eroded by B1, intersects A complement eroded by B2 (two different structuring elements: B1 , B2) Roger S. Gaborski 47

  47. Hit or Miss Example Find cross shape pixel configuration: MATLAB Function: C = bwhitmiss(A, B1, B2) Roger S. Gaborski 48

  48. Original Image A and B1 A eroded by B1 Complement of Original Image and B2 Erosion of A complement And B2 Intersection of eroded images Roger S. Gaborski 49 From: Digital Image Processing, Gonzalez,Woods And Eddins

  49. Original Image A and B1 A eroded by B1 Complement of Original Image and B2 Erosion of A complement And B2 Intersection of eroded images Roger S. Gaborski 50 From: Digital Image Processing, Gonzalez,Woods And Eddins