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3-D Computater Vision CSc 83020

3-D Computater Vision CSc 83020. Revisit filtering (Gaussian and Median) Introduction to edge detection. Linear Filters. Given an image In ( x , y ) generate a new image Out ( x , y ):

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3-D Computater Vision CSc 83020

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  1. 3-D Computater VisionCSc 83020 • Revisit filtering (Gaussian and Median) • Introduction to edge detection 3-D Computer Vision CSc83020 / Ioannis Stamos

  2. Linear Filters • Given an image In(x,y) generate anew image Out(x,y): • For each pixel (x,y)Out(x,y) is a linear combination of pixelsin the neighborhood of In(x,y) • This algorithm is • Linear in input intensity • Shift invariant 3-D Computer Vision CSc83020 / Ioannis Stamos

  3. Discrete Convolution • This is the discrete analogue of convolution • The pattern of weights is called the “kernel”of the filter • Will be useful in smoothing, edge detection 3-D Computer Vision CSc83020 / Ioannis Stamos

  4. Computing Convolutions • What happens near edges of image? • Ignore (Out is smaller than In) • Pad with zeros (edges get dark) • Replicate edge pixels • Wrap around • Reflect • Change filter 3-D Computer Vision CSc83020 / Ioannis Stamos

  5. Example: Smoothing Original: Mandrill Smoothed withGaussian kernel 3-D Computer Vision CSc83020 / Ioannis Stamos

  6. Gaussian Filters • One-dimensional Gaussian • Two-dimensional Gaussian 3-D Computer Vision CSc83020 / Ioannis Stamos

  7. Gaussian Filters 3-D Computer Vision CSc83020 / Ioannis Stamos

  8. Gaussian Filters 3-D Computer Vision CSc83020 / Ioannis Stamos

  9. Gaussian Filters • Gaussians are used because: • Smooth • Decay to zero rapidly • Simple analytic formula • Limit of applying multiple filters is Gaussian(Central limit theorem) • Separable: G2(x,y) = G1(x) G1(y) 3-D Computer Vision CSc83020 / Ioannis Stamos

  10. Size of the mask 3-D Computer Vision CSc83020 / Ioannis Stamos

  11. Edges & Edge Detection • What are Edges? • Theory of Edge Detection. • Edge Operators (Convolution Masks) • Edge Detection in the Brain? • Edge Detection using Resolution Pyramids 3-D Computer Vision CSc83020 / Ioannis Stamos

  12. Edges 3-D Computer Vision CSc83020 / Ioannis Stamos

  13. What are Edges? Rapid Changes of intensity in small region 3-D Computer Vision CSc83020 / Ioannis Stamos

  14. What are Edges? Surface-Normal discontinuity Depth discontinuity Surface-Reflectance Discontinuity Illumination Discontinuity Rapid Changes of intensity in small region 3-D Computer Vision CSc83020 / Ioannis Stamos

  15. Local Edge Detection 3-D Computer Vision CSc83020 / Ioannis Stamos

  16. Edge easy to find What is an Edge? 3-D Computer Vision CSc83020 / Ioannis Stamos

  17. What is an Edge? Where is edge? Single pixel wide or multiple pixels? 3-D Computer Vision CSc83020 / Ioannis Stamos

  18. What is an Edge? Noise: have to distinguish noise from actual edge 3-D Computer Vision CSc83020 / Ioannis Stamos

  19. What is an Edge? Is this one edge or two? 3-D Computer Vision CSc83020 / Ioannis Stamos

  20. What is an Edge? Texture discontinuity 3-D Computer Vision CSc83020 / Ioannis Stamos

  21. Local Edge Detection 3-D Computer Vision CSc83020 / Ioannis Stamos

  22. Edge Types Ideal Step Edges Ideal Ridge Edges Ideal Roof Edges

  23. Real Edges I x Problems: Noisy Images Discrete Images 3-D Computer Vision CSc83020 / Ioannis Stamos

  24. Real Edges We want an Edge Operator that produces: Edge Magnitude (strength) Edge direction Edge normal Edge position/center High detection rate & good localization 3-D Computer Vision CSc83020 / Ioannis Stamos

  25. The 3 steps of Edge Detection • Noise smoothing • Edge Enhancement • Edge Localization • Nonmaximum suppression • Thresholding 3-D Computer Vision CSc83020 / Ioannis Stamos

  26. Theory of Edge Detection Unit Step Function: y B1,L(x,y)>0 t B2,L(x,y)<0 x 3-D Computer Vision CSc83020 / Ioannis Stamos

  27. Theory of Edge Detection Unit Step Function: y B1,L(x,y)>0 t B2,L(x,y)<0 x Ideal Edge: Image Intensity (Brightness): 3-D Computer Vision CSc83020 / Ioannis Stamos

  28. Theory of Edge Detection Partial Derivatives: y B1,L(x,y)>0 t B2,L(x,y)<0 Directional! x 3-D Computer Vision CSc83020 / Ioannis Stamos

  29. Theory of Edge Detection y B1,L(x,y)>0 t B2,L(x,y)<0 x Squared Gradient: Edge Magnitude Edge Orientation Rotationally Symmetric, Non-Linear 3-D Computer Vision CSc83020 / Ioannis Stamos

  30. Theory of Edge Detection Laplacian: y B1,L(x,y)>0 t B2,L(x,y)<0 x (Rotationally Symmetric & Linear) I x x Zero Crossing

  31. Difference Operators Ii,j+1 Ii+1,j+1 ε Ii,j Ii+1,j Finite Difference Approximations 3-D Computer Vision CSc83020 / Ioannis Stamos

  32. Squared Gradient y x 3-D Computer Vision CSc83020 / Ioannis Stamos

  33. Squared Gradient [Roberts ’65] if threshold then we have an edge 3-D Computer Vision CSc83020 / Ioannis Stamos

  34. Squared Gradient– Sobel Mean filter convolved with first derivative filter 3-D Computer Vision CSc83020 / Ioannis Stamos

  35. Examples First derivative Sobel operator 3-D Computer Vision CSc83020 / Ioannis Stamos

  36. Second Derivative Edge occurs at the zero-crossing of the second derivative 3-D Computer Vision CSc83020 / Ioannis Stamos

  37. Laplacian • Rotationally symmetric • Linear computation (convolution) 3-D Computer Vision CSc83020 / Ioannis Stamos

  38. Discrete Laplacian Ii,j+1 Ii+1,j+1 Ii-1,j+1 Ii,j Ii+1,j Ii-1,j Ii-1,j-1 Ii,j-1 Ii+1,j-1 Finite Difference Approximations 3-D Computer Vision CSc83020 / Ioannis Stamos

  39. Discrete Laplacian More accurate • Rotationally symmetric • Linear computation (convolution) 3-D Computer Vision CSc83020 / Ioannis Stamos

  40. Discrete Laplacian Laplacian of an image 3-D Computer Vision CSc83020 / Ioannis Stamos

  41. Discrete Laplacian Laplacian is sensitive to noise First smooth image with Gaussian 3-D Computer Vision CSc83020 / Ioannis Stamos

  42. From Forsyth & Ponce. 3-D Computer Vision CSc83020 / Ioannis Stamos

  43. From Shree Nayar’s notes. 3-D Computer Vision CSc83020 / Ioannis Stamos

  44. Discrete Laplacian w/ Smoothing 3-D Computer Vision CSc83020 / Ioannis Stamos

  45. From Shree Nayar’s notes. 3-D Computer Vision CSc83020 / Ioannis Stamos

  46. Difference Operators – Second Derivative 3-D Computer Vision CSc83020 / Ioannis Stamos

  47. From Forsyth & Ponce. 3-D Computer Vision CSc83020 / Ioannis Stamos

  48. Edge Detection – Human Vision LoG convolution in the brain – biological evidence! Flipped LoG LoG 3-D Computer Vision CSc83020 / Ioannis Stamos

  49. Image Resolution Pyramids Can save computations. Consolidation: Average pixels at one level to find value at higher level. Template Matching: Find match in COARSE resolution. Then move to FINER resolution.

  50. From Forsyth & Ponce. 3-D Computer Vision CSc83020 / Ioannis Stamos

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