1. 1 Model Fitting Hao Jiang Computer Science Department Oct 8, 2009

2. Hough transform for circles  For a fixed radius r, unknown gradient direction Circle: center (a,b) and radius r Image space Hough space Source: K. Grauman

3. Hough transform for circles  For a fixed radius r, unknown gradient direction Circle: center (a,b) and radius r Image space Hough space Intersecti on: most votes for center occur here. Source: K. Grauman

4. Hough transform for circles  For an unknown radius r, unknown gradient direction Circle: center (a,b) and radius r Hough space Image space b a r Source: K. Grauman

5. Hough transform for circles  For an unknown radius r, unknown gradient direction Circle: center (a,b) and radius r Hough space Image space b a r Source: K. Grauman

6. Hough transform for circles  For an unknown radius r, known gradient direction Circle: center (a,b) and radius r Hough spaceImage space θ x Source: K. Grauman

7. Hough transform for circles For every edge pixel (x,y) : For each possible radius value r: For each possible gradient direction θ: // or use estimated gradient a = x – r cos(θ) b = y + r sin(θ) H[a,b,r] += 1 end Source: K. Grauman

8. Example: detecting circles with Hough Crosshair indicates results of Hough transform, bounding box found via motion differencing. Source: K. Grauman

9. Original Edges Example: detecting circles with Hough Votes: Penny Note: a different Hough transform (with separate accumulators) was used for each circle radius (quarters vs. penny). Source: K. Grauman

10. Original Edges Example: detecting circles with Hough Votes: Quarter Coin finding sample images from: Vivek Kwatra Source: K. Grauman

11. General Hough Transform  Hough transform can also be used to detection arbitrary shaped object. 11 TemplateTarget

12. General Hough Transform  Hough transform can also be used to detection arbitrary shaped object. 12 TemplateTarget

13. General Hough Transform  Hough transform can also be used to detection arbitrary shaped object. 13 TemplateTarget

14. General Hough Transform  Hough transform can also be used to detection arbitrary shaped object. 14 TemplateTarget

15. General Hough Transform  Rotation Invariance 15

16. Example model shape Source: L. Lazebnik Say we’ve already stored a table of displacement vectors as a function of edge orientation for this model shape.

17. Example displacement vectors for model points Now we want to look at some edge points detected in a new image, and vote on the position of that shape.

18. Example range of voting locations for test point

19. Example range of voting locations for test point

20. Example votes for points with θ =

21. Example displacement vectors for model points

22. Example range of voting locations for test point

23. votes for points with θ = Example

24. Application in recognition Instead of indexing displacements by gradient orientation, index by “visual codeword” B. Leibe, A. Leonardis, and B. Schiele, Combined Object Categorization and Segmentation with an Implicit Shape ModelCombined Object Categorization and Segmentation with an Implicit Shape Model, ECCV Workshop on Statistical Learning in Computer Vision 2004 training image visual codeword with displacement vectors Source: L. Lazebnik

25. Application in recognition Instead of indexing displacements by gradient orientation, index by “visual codeword” test image Source: L. Lazebnik

26. RANSAC  RANSAC – Random Sampling Consensus  Procedure:  Randomly generate a hypothesis  Test the hypothesis  Repeat for large enough times and choose the best one 26

27. Line Detection 27

28. Line Detection 28

29. Line Detection 29

30. Line Detection 30 Count “inliers”: 7

31. Line Detection 31 Inlier number: 3

32. Line Detection 32 After many trails, we decide this is the best line.

33. Probability of Success  What’s the success rate of RANSAC given a fixed number of validations?  Given a success rate, how to choose the minimum number of validations? 33

34. Object Detection Using RANSAC 34 Template Target

35. Scale and Rotation Invariant Feature  SIFT (D. Lowe, UBC) 35

36. Stable Feature 36

37. Stable Feature 37 Local max/min point’s values are stable when the scale changes

38. SIFT 38 Filtering the image using filters at different scales. (for example using Gaussian filter)

39. Difference of Gaussian 39

40. SIFT Feature Points 40 (b) Shows the points at the local max/min of DOG scale space for the image in (a).