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

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

3. 3-D Computer Vision CSc83020 / Ioannis Stamos 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

4. 3-D Computer Vision CSc83020 / Ioannis Stamos 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

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

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

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

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

9. 3-D Computer Vision CSc83020 / Ioannis Stamos 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: G 2 (x,y) = G 1 (x) G 1 (y)

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

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

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

13. What are Edges? Rapid Changes of intensity in small region

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

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

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

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

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

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

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

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

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

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

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

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

26. 3-D Computer Vision CSc83020 / Ioannis Stamos Theory of Edge Detection x yB1,L(x,y)>0 B2,L(x,y)<0 t Unit Step Function:

27. 3-D Computer Vision CSc83020 / Ioannis Stamos Theory of Edge Detection x yB1,L(x,y)>0 B2,L(x,y)<0 t Unit Step Function: Ideal Edge: Image Intensity (Brightness):

28. 3-D Computer Vision CSc83020 / Ioannis Stamos Theory of Edge Detection x yB1,L(x,y)>0 B2,L(x,y)<0 t Partial Derivatives: Directional!

29. 3-D Computer Vision CSc83020 / Ioannis Stamos Theory of Edge Detection x yB1,L(x,y)>0 B2,L(x,y)<0 t Rotationally Symmetric, Non-Linear Edge Magnitude Edge Orientation Squared Gradient:

30. Theory of Edge Detection x yB1,L(x,y)>0 B2,L(x,y)<0 t Laplacian: (Rotationally Symmetric & Linear) I xx Zero Crossing

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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