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Introduction to Computer Vision

Lecture 07 Roger S. Gaborski. Introduction to Computer Vision. Roger S. Gaborski. 1. Simple First Derivative Approximation. Difference . x. =. Smoothing. Rotate Filter. Sensitive to edges at different orientations. Sobel Filter. Consider unequal weights for smoothing operation. = Sy.

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Introduction to Computer Vision

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

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

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

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

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

  6. build1.jpg Roger S. GaborskiRoger S. Gaborski

  7. Edge image Roger S. GaborskiRoger S. Gaborski

  8. ~ Operator Roger S. GaborskiRoger S. Gaborski

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

  10. 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

  11. 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

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

  13. 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

  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. GaborskiRoger S. Gaborski

  15. 7x7 Gaussian Filter Roger S. GaborskiRoger S. Gaborski

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

  17. 31x31Gaussian, sigma = 3 Roger S. GaborskiRoger S. Gaborski

  18. 63x63 Gaussian, sigma = 10 Roger S. GaborskiRoger S. Gaborski

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

  20. 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

  21. [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

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

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

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

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

  26. 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

  27. 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

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

  29. 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

  30. 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

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

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

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

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

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

  36. 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

  37. 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

  38. 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

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

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

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

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

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

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

  45. t = .500 Sigma = 1 Sigma = 3 Roger S. GaborskiRoger S. Gaborski

  46. t = 1.500 Sigma = 1.0 Sigma = 3.0 Roger S. GaborskiRoger S. Gaborski

  47. DoG • Difference of Gaussians is an approximation to Laplacian of Gaussian Roger S. GaborskiRoger S. Gaborski

  48. Comparison of Canny and LoG

  49. LoG St dev = 2 Canny, Std dev = 1 Roger S. GaborskiRoger S. Gaborski

  50. The fern image [g,t]=edge(fernGray,'canny',[ ],2); The structure of the image affects the performance of edge detection Roger S. GaborskiRoger S. Gaborski

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