Introduction to Computer Vision

1. Lecture 07 Roger S. Gaborski Introduction to Computer Vision Roger S. GaborskiRoger S. Gaborski Roger S. Gaborski 1

2. Linear Filtering in MATLAB g = imfilter(f, w, filtering mode, boundary, size) • filters the imput image f with the filter mask w. • f is input image. It can be of any class (logical/numeric) and dimension. • g is output image • filter mode: - 'corr' : correlation, and default mode - 'conv' : convolution Roger S. Gaborski

3. Parameters • g = imfilter(f, w, filtering mode, boundary, size) Boundary options - X pad boundary with value X. Default X = 0. - 'symmetric' symmetric padding - 'replicate' replicate padding - 'circular' circular padding Size options - 'same' g is the same size of f (default mode) - 'full' g is full filtered by w, so size of g is increased Roger S. Gaborski

4. MATLAB function for filtering: imfilter • g = imfilter(f, w, ‘replicate’) • Correlation is the default filtering mode. • If filters are pre-rotated 180 degrees, can simply use default(corr) for convolution • If filter is symmetric, doesn’t matter Roger S. Gaborski

5. Simple First Derivative Approximation Difference x = Smoothing Roger S. GaborskiRoger S. Gaborski

6. Rotate Filter Sensitive to edges at different orientations Roger S. GaborskiRoger S. Gaborski

7. Sobel Filter • Consider unequal weights for smoothing operation = Sy x = Roger S. GaborskiRoger S. Gaborski

8. Sobel Filter • Consider unequal weights for smoothing operation = Sx x = Roger S. GaborskiRoger S. Gaborski

9. House Image >> I = imread('house1.jpg'); >> figure, imshow(I) >> Jg = rgb2gray(J); >> Jg = im2double(Jg); >> figure, imshow(Jg) Roger S. GaborskiRoger S. Gaborski

10. Horizontal Edge Detection >> f = fspecial('sobel') f = 1 2 1 0 0 0 -1 -2 -1 >> imEdge = imfilter(Jg, f ); >> figure, imshow(imEdge,[ ]); Roger S. GaborskiRoger S. Gaborski

11. >> figure, imshow(abs(imEdge),[]) Roger S. GaborskiRoger S. Gaborski

12. >> figure, hist(abs(imEdge(:)),100) Roger S. GaborskiRoger S. Gaborski

13. figure, imshow(imEdge>.5) Roger S. GaborskiRoger S. Gaborski

14. Vertical Edges >> fv = f' fv = 1 0 -1 2 0 -2 1 0 -1 >> imEdgeV = imfilter(Jg, fv ); >> figure, imshow(imEdgeV,[]); Roger S. GaborskiRoger S. Gaborski

15. Both Vertical and Horizontal Edges figure, subplot(1,2,1),imshow(abs(Eh)>.5) >> subplot(1,2,2),imshow(abs(Ev)>.5) Roger S. GaborskiRoger S. Gaborski

16. figure, imshow((abs(Eh)+abs(Ev))/2,[]) Roger S. GaborskiRoger S. Gaborski

17. build1.jpg Roger S. GaborskiRoger S. Gaborski

18. Edge image Roger S. GaborskiRoger S. Gaborski

19. ~ Operator Roger S. GaborskiRoger S. Gaborski

20. Is there a better way to remove noise than the simple Sobel approach? Roger S. GaborskiRoger S. Gaborski

21. Canny • Image is smoothed with 2D Gaussian Function • Canny uses two thresholds • One threshold detects strong edges • Second threshold detects weak edges connected to strong edges Roger S. GaborskiRoger S. Gaborski

22. RECALL: 2D Gaussian Distribution • The two-dimensional Gaussian distribution is defined by: • From this distribution, can generate smoothing masks whose width depends upon the standard deviation, s: Roger S. GaborskiRoger S. Gaborski

23. Sigma Determines Spread of Filter Variance, s2 = .25 Variance, s2 = 4.0 Roger S. GaborskiRoger S. Gaborski

24. i2 + j2 i2 + j2 W(i,j) = exp (- ) W(i,j) = k * exp (- ) 2 s2 2 s2 k Creating Gaussian Kernels • The mask weights are evaluated from the Gaussian distribution: • This can be rewritten as: Roger S. GaborskiRoger S. Gaborski

25. j -3 -2 -1 0 1 2 3 -3 -2 -1 i 0 1 2 3 Example 2 • Choose s = 2. and n = 7, then: Roger S. GaborskiRoger S. Gaborski

26. 7x7 Gaussian Filter Roger S. GaborskiRoger S. Gaborski

27. Building1 gray level image Roger S. GaborskiRoger S. Gaborski

28. 31x31Gaussian, sigma = 3 Roger S. GaborskiRoger S. Gaborski

29. 63x63 Gaussian, sigma = 10 Roger S. GaborskiRoger S. Gaborski

30. Canny Edge Detector Roger S. GaborskiRoger S. Gaborski

31. Implement Canny usingMATLAB edge Function EDGE Find edges in intensity image. EDGE takes an intensity or a binary image I as its input, and returns a binary image BW of the same size as I, with 1's where the function finds edges in I and 0's elsewhere. EDGE supports six different edge-finding methods: The Sobel method finds edges using the Sobel approximation to the derivative. The Prewitt method finds edges using the Prewitt approximation to the derivative. The Roberts method finds edges using the Roberts approximation to the derivative. The Laplacian of Gaussian method finds edges by looking for zero crossings after filtering I with a Laplacian of Gaussian filter. The zero-cross method finds edges by looking for zero crossings after filtering I with a filter you specify. The Canny method finds edges by looking for local maxima of the gradient of I. The gradient is calculated using the derivative of a Gaussian filter. The method uses two thresholds, to detect strong and weak edges, and includes the weak edges in the output only if they are connected to strong edges. This method is therefore less likely than the others to be "fooled" by noise, and more likely to detect true weak edges. Roger S. GaborskiRoger S. Gaborski

32. [g,t]=edge(im,'canny',[ ],.5) [ ] : Edge function determines the two thresholds CHANGING SIGMA OF GAUSSIAN, in this example = .5 Roger S. GaborskiRoger S. Gaborski

33. [g,t]=edge(im,'canny',[ ],1) CHANGING SIGMA OF GAUSSIAN Roger S. GaborskiRoger S. Gaborski

34. [g,t]=edge(im,'canny',[ ],2) CHANGING SIGMA OF GAUSSIAN Roger S. GaborskiRoger S. Gaborski

35. [g,t]=edge(im,'canny',[ ],3) CHANGING SIGMA OF GAUSSIAN Roger S. GaborskiRoger S. Gaborski

36. [g,t]=edge(im,'canny',[ ],5) CHANGING SIGMA OF GAUSSIAN Roger S. GaborskiRoger S. Gaborski

37. Second Derivative Edge Detection Methods • Second derivative is noisy • First smooth the image, then apply second derivative • Consider • Effect of smoothing filter. • Gaussian: Roger S. GaborskiRoger S. Gaborski

38. RECALL: Second Derivative Approximation • Discrete version of 2nd Partial Derivation of f(x,y) in x direction is found by taking the difference of Eq1 and Eq2: • 2 f(x,y) /  x2 = Eq1-Eq2 = 2f(x,y)-f(x-1,y)-f(x+1,y) In y direction: • 2 f(x,y) /  y2 = Eq3-Eq4 = 2f(x,y)-f(x,y-1)-f(x,y+1) Roger S. GaborskiRoger S. Gaborski

39. Derivative Approximation for Laplacian Roger S. GaborskiRoger S. Gaborski

40. Laplacian • Independent of edge orientation • Combine 2 f(x,y)/ x2 and 2 f(x,y)/ y2 =4 f(x,y) - f(x-1,y) – f(x+1,y) – f(x,y-1) – f(x,y+1) Roger S. GaborskiRoger S. Gaborski

41. Laplacian of the Gaussian is a circularly symmetric operator. Second derivative is linear operation Therefore: convolving an image with 2 G(x,y) is the same as first convolving first with smoothing filter (Gaussian) then computing Laplacian of result. Edge are location of zero crossings LoG Roger S. GaborskiRoger S. Gaborski

42. LoG Also called the Mexican hat operator. Roger S. GaborskiRoger S. Gaborski

43. s2 Controls of the Size of the Filter s2 = 0.5 s2 = 2.0 Roger S. GaborskiRoger S. Gaborski

44. Human Visual System Receptive Field Approximation 17 x 17 5x5 Roger S. GaborskiRoger S. Gaborski

45. LoG (7x7, sigma = 2) Roger S. GaborskiRoger S. Gaborski

46. LoG (15x15, sigma = 4) Roger S. GaborskiRoger S. Gaborski

47. Summarizing: Laplacian of Gaussian (LoG) • Gaussian function: h(r) = -exp(-r2/22) • Applying the Gaussian has a smoothing or blurring effect • Blurring depends on sigma • THEN Laplacian of Gaussian (Second Derivative) Roger S. GaborskiRoger S. Gaborski

48. MATLAB edge Function EDGE Find edges in intensity image. EDGE takes an intensity or a binary image I as its input, and returns a binary image BW of the same size as I, with 1's where the function finds edges in I and 0's elsewhere. EDGE supports six different edge-finding methods: The Sobel method finds edges using the Sobel approximation to the derivative. The Prewitt method finds edges using the Prewitt approximation to the derivative. The Roberts method finds edges using the Roberts approximation to the derivative. The Laplacian of Gaussian method finds edges by looking for zero crossings after filtering I with a Laplacian of Gaussian filter. The zero-cross method finds edges by looking for zero crossings after filtering I with a filter you specify. The Canny method finds edges by looking for local maxima of the gradient of I. The gradient is calculated using the derivative of a Gaussian filter. The method uses two thresholds, to detect strong and weak edges, and includes the weak edges in the output only if they are connected to strong edges. This method is therefore less likely than the others to be "fooled" by noise, and more likely to detect true weak edges. Roger S. GaborskiRoger S. Gaborski

49. Implementation: Laplacian of Gaussian (LoG) • Syntax: [g] = edge(f, ‘log’, T, sigma); Ignores edges that are not stronger than T If T not provided, MATLAB automatically chooses T Roger S. GaborskiRoger S. Gaborski

50. Building What’s important?? Roger S. GaborskiRoger S. Gaborski