Introduction to Computer Vision

1. Introduction to Computer Vision Lecture 8 Dr. Roger S. Gaborski Roger S. Gaborski

2. Updates • Project Page • Image Resizing by Image Carving http://www.youtube.com/watch?v=6NcIJXTlugc Roger S. Gaborski

3. GOAL: Locate the Fern Roger S. Gaborski

4. RECALL: MATLAB’s 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. Sobel Prewitt Roberts Laplacian of Gaussian (LoG) Zero Crossing Canny Roger S. Gaborski

5. edge • edge function • [g,t] = edge( image, ‘prewitt’, T, dir ); Roger S. Gaborski

6. Prewitt [g,t] = edge( image, ‘prewitt’ ); Automatic threshold = .1088, both horizontal and vertical edges Roger S. Gaborski

7. Prewitt – Threshold = .05 Threshold = .05 Roger S. Gaborski

8. Prewitt – Threshold =.15 Threshold = .15 Roger S. Gaborski

9. Smoothing • Remove noise • Remove small details • Smoothing Kernel: a= 1/9 a a a a a a a a a • Increase center value to put more weight on the • center pixel Roger S. Gaborski

10. Consider a 2 D Gaussian for Smoothing Roger S. Gaborski

11. 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. Gaborski

12. 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. Gaborski

13. i2 + j2 W(i,j) = exp (- ) 2 s2 k Creating Gaussian Kernels • This can now be evaluated over a window of size nxn to obtain a kernel in which the (0,0) value is 1. • k is a scaling constant Roger S. Gaborski

14. 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. Gaborski

15. 7x7 Gaussian Filter Roger S. Gaborski

16. Key Point – if you were using a Gaussian function for smoothing (noise reduction) the value of will sigma determine how many points will be used in the smoothing operation Roger S. Gaborski

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

18. imGray im = imread('IMGP1579.JPG'); imGray = rgb2gray(im); imGray = mat2gray(imGray, [0 255]); Roger S. Gaborski

19. 31x31 Gaussian, Sigma = 4 Roger S. Gaborski

20. 31x31 Gaussian, Sigma = 4Profile Roger S. Gaborski

21. Fern 31x31 Gaussian, Sigma=4 Roger S. Gaborski

22. 63x63 Gaussian, Sigma =10 Roger S. Gaborski

23. 63x63 Gaussian, Sigma = 10Profile Roger S. Gaborski

24. Fern 63x63 Gaussian, Sigma=10 Roger S. Gaborski

25. RECALL: Laplacian • Independent of edge orientation (finds edges in all orientations) • 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) • Second derivatives are sensitive to noise Roger S. Gaborski

26. Filter image with Gaussian to reduce noise Laplacian of the Gaussian Ñ2G is a circularly symmetric operator. LoG Roger S. Gaborski

27. LoG Also called the Mexican hat operator. Roger S. Gaborski

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

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

30. >> %H = FSPECIAL('log',HSIZE,SIGMA) >> LoG31=fspecial('log',31,4); >> imLoG31=conv2(imGray,LoG31); >> figure, imshow(imLoG31,[ ]),title('imLog31') Roger S. Gaborski

31. LoG (15x15, Sigma=1) Roger S. Gaborski

32. LoG (7x7, Sigma=1) Roger S. Gaborski

33. LoG (31x31, Sigma=4) Roger S. Gaborski

34. LoG (63x63, Sigma=10) Roger S. Gaborski

35. Edge Detection: abs(imLog7)>.1 Roger S. Gaborski

36. Building1 gray level image Roger S. Gaborski

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

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

39. 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. Gaborski

40. Laplacian of Gaussian (LoG) • 2 h(r) = -[(r2 - 2)/ 4] exp-(r2/2 2) • Second derivative is linear operation Therefore: convolving an image with 2 h(r) is the same as first convolving first with smoothing filter (Gaussian) then computing Laplacian of result. • Edge are location of zero crossings Roger S. Gaborski

41. 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. Gaborski

42. Building What’s important?? Roger S. Gaborski

43. [g,t]=edge(im,'log',[ ],1); Roger S. Gaborski

44. [g,t]=edge(im,'log',[ ],2); Roger S. Gaborski

45. [g,t]=edge(im,'log',[ ],3) Roger S. Gaborski

46. Exploring the LoG Parameter Space [g, t] = edge(f, 'log',[],2); Default threshold, t= .918, sigma = 2 WHAT IF WE CHANGE SIGMA?? Roger S. Gaborski

47. t= .918 Sigma = 1 Sigma = 3 (more smoothing) Roger S. Gaborski

48. t = .500 Sigma = 1 Sigma = 3 Roger S. Gaborski

49. t = 1.500 Sigma = 1.0 Sigma = 3.0 Roger S. Gaborski

50. DoG • NOTE: Difference of Gaussians is an approximation to Laplacian of Gaussian Roger S. Gaborski