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Linear Solution to Scale and Rotation Invariant Object Matching

Linear Solution to Scale and Rotation Invariant Object Matching. Professor: 王聖智 教授 Student : 周 節. Outline. Introduction Method Result Conclusion Reference. Problem. Template image Target image. Problem. Template image and mesh. Target image.

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Linear Solution to Scale and Rotation Invariant Object Matching

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  1. Linear Solution to Scale and Rotation Invariant Object Matching Professor:王聖智教授 Student :周 節

  2. Outline • Introduction • Method • Result • Conclusion • Reference

  3. Problem Template image Target image

  4. Problem Template image and mesh Target image Matching Target Mesh

  5. Challenges Template Image Target Images

  6. Challenges Template Image Target Images

  7. Related Works :w • Hough Transform and RANSAC • Graph matching • Dynamic Programming • Max flow – min cut [Ishikawa 2000, Roy 98] • Greedy schemes (ICM [Besag 86], Relaxation Labeling [Rosenfeld 76]) • Back tracking with heuristics [Grimson 1988] • Graph Cuts [Boykov & Zabih 2001] • Belief Propagation [Pearl 88, Weiss 2001] • Convex approximation [Berg 2005, Jiang 2007, etc.]

  8. Method • A linear method to optimize the matching from template to target • The linear approximation is simplified so that its size is largely decoupled from the number of target candidate features points • Successive refinement for accurate matching

  9. Method • A linear method to optimize the matching from template to target • The linear approximation is simplified so that its size is largely decoupled from the number of target candidate features points • Successive refinement for accurate matching

  10. The Optimization Problem

  11. The Optimization Problem Matching cost Pairwise consistency

  12. In Compact Matrix Form Matching cost Pairwise consistency

  13. In Compact Matrix Form Matching cost Pairwise consistency We will turn the terms in circles to linear approximations

  14. The L1 Norm Linearization • It is well known that L1 norm is “linear” min |x| min (y + z) Subject to x = y - z y, z >= 0

  15. The L1 Norm Linearization • It is well known that L1 norm is “linear” • By using two auxiliary matrices Y and Z, min |x| min (y + z) Subject to x = y - z y, z >= 0 | EMR – sEXT | 1’ne( Y + Z ) 12 subject to Y – Z = EMR – sEXT Y, Z >= 0 Consistency term

  16. Linearize the Scale Term For any given s Cut into Ns part 0 <S1<. . .< Sns

  17. Linearize the Scale Term For any given s Cut into Ns part 0 <S1<. . .< Sns For any given X X1, X2…Xns

  18. Linearize the Scale Term For any given s Cut into Ns part 0 <S1<. . .< Sns For any given X X1, X2…Xns

  19. Linearize the Scale Term For any given s Cut into Ns part 0 <S1<. . .< Sns For any given X X1, X2…Xns

  20. Linearize the Scale Term All sites select the same scale

  21. Linearize Rotation Matrix In graphics:

  22. The Linear Optimization Target point estimation: T* = XT

  23. Method • A linear method to optimize the matching from template to target • The linear approximation is simplified so that its size is largely decoupled from the number of target candidate features points • Successive refinement for accurate matching

  24. Convex optimization problem Reference: http://www.stanford.edu/class/ee364a/lectures/problems.pdf

  25. The Lower Convex Hull Property • Cost surfaces can be replaced by their lower convex hulls without changing the LP solution A cost “surface” for one point on the template The lower convex hull

  26. Removing Unnecessary Variables • Only the variablesthat correspond to the lower convex hull vertices need to be involved in the optimization matching cost matching cost # of original target candidates: 2500 # of effective target candidates: 19

  27. Complexity of the LP 10,000 target candidates 1 million target candidates 28 effective target candidates 1034 effective target candidates The size of the LP is largely decoupled from the number of target candidates

  28. Method • A linear method to optimize the matching from template to target • The linear approximation is simplified so that its size is largely decoupled from the number of target candidate features points • Successive refinement for accurate matching

  29. Successive Refinement

  30. Result

  31. Matching Objects in Real Images

  32. Result Videos

  33. Statistics

  34. Results Comparison

  35. Accuracy and Efficiency

  36. Experiments on Ground Truth Data The proposed method Other methods Error distributions for fish dataset

  37. Experiments on Ground Truth Data The proposed method Other methods Error distributions for random dot dataset

  38. Conclusion • A linear method to solve scale and rotation invariant matching problems accurately and efficiently • The proposed method is flexible and can be used to match images using different features • It is useful for many applications including object detection, tracking and activity recognition

  39. Reference • Hao Jiang and Stella X. Yu, “Linear Solution to Scale and Rotation Invariant Object Matching”, IEEE Conference on Computer Vision and Pattern Recognition 2009 (CVPR'09) • http://www.stanford.edu/class/ee364a/lectures.html

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