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Edge Detection in Image Processing: Theory, Techniques, and Applications

Explore the theory, techniques, and applications of edge detection in image processing, including linear filters, Gaussian filters, and Laplacian operators. Learn about detecting edges, smoothing images, and optimizing edge detection algorithms for various applications.

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Edge Detection in Image Processing: Theory, Techniques, and Applications

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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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