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Scale and Rotation Invariant Matching Using Linearly Augmented Tree

This research proposes the Linearly Augmented Tree (LAT) method for efficient and accurate scale and rotation invariant matching. LAT incorporates global constraints and works well with weak features and large deformations. It does not require an upper bound for the scale and can enhance object detection and pose estimation methods.

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Scale and Rotation Invariant Matching Using Linearly Augmented Tree

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  1. Scale and Rotation Invariant Matching Using Linearly Augmented Tree Hao Jiang Boston College Tai-peng Tian, Stan Sclaroff Boston University

  2. Scale and Rotation Invariant Matching

  3. Previous Methods • Hough Transform (Duda & Hart) and RANSAC (Fischler and Bolles) • Dynamic programming (Felzenszwalb and Huttenlocher 05) • Loopy belief propagation (Weiss and Freeman 01) • Tree-reweighted message passing (Kolmogorov 06) • Primal-dual methods (Komodakis and Tziritas 07) • Dual decomposition (Komodakis, Paragios and Tziritas 11, Torresani, Kolmogorov and Rother 08) • Successive convexification (Jiang 2009, Li, Kim, Huang and He 2010)

  4. Unsolved Issue • How to find the optimal rotation angle and scale especially if the ranges are unknown? Quantizing rotation angle and scale

  5. The Model We all Want to Have

  6. In Reality We Need to Use … Hyperedge Non-tree edge

  7. Linearly Augmented Tree (LAT) Linear non-tree constraints Any tree constraints LAT works on continuous scale and rotation and non-tree structure.

  8. Optimizing Invariant Matching cost(p,f(p)) Total local feature matching cost p f(p) … … q f(q) cost(q,f(q))

  9. Optimizing Invariant Matching In matrix form: X Binary assignment Matrix C Local matching cost matrix

  10. Optimizing Invariant Matching Rotation and scaling consistency Model tree edges Target

  11. Optimizing Invariant Matching Yp,q Pairwise assignment matrix for site pair (p,q) s0, u0 = sin(θ0), v0 = cos(θ0) Θp,q Sp,q Rotation angle matrix Scale matrix

  12. Optimizing Invariant Matching Other linear global terms such as area constraints or global affine constraints. Area scaling is

  13. The Mixed Integer Optimization Unary matching cost Rotation consistency Scaling term g(X) Other global terms Subject To: Constraints on binary matrices X, Y, and continuous variables u0, v0 and s0.

  14. Linear Relaxation

  15. Special Structure Objective function “Hard” constraints X, Y Auxiliary variables Easy Ones

  16. The Solution Space Optimal solution Solutions feasible to the “simple” problem. Solutions feasible to “hard” constraints.

  17. Column Generation Proposals Two initial proposals and the current best estimate.

  18. Column Generation Proposals Few extreme points (proposals) can be used to obtain the solution, and they can be generated iteratively.

  19. Decompose into Dynamic Programming Create the initial trellises, and find first 2 proposals k=2 Find out how to update the tree Yes Gain > 0 Done Update the trees Dynamic Programming and generate new proposal (k+1)

  20. An Example Template Image

  21. An Example Template

  22. An Example Target Image

  23. An Example

  24. Another Example Template

  25. Another Example Target Image

  26. Another Example

  27. Complexity Comparison Direct solution

  28. SIFT Matching Results Detection Rate

  29. Match Unreliable Regions Detection Rate

  30. Matching Unreliable Regions Detection Rate

  31. More Tests

  32. Test on Ground Truth Data

  33. Summary • LAT method can incorporate global constraints and can be efficiently solved. • It works even with very weak features and large deformations in scale and rotation invariant matching. • It works on continuous scale and rotation and does not need an upper bound for the scale. • The decomposition framework would be useful to enhance the widely applied tree methods in object detection, pose estimation and etc.

  34. The End

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