An Optimization Approach to Improving Collections of Shape Maps

1. An Optimization Approach to Improving Collections of Shape Maps Andy Nguyen, MirelaBen-Chen, KatarzynaWelnicka, Yinyu Ye, Leonidas Guibas Computer Science Dept. Stanford University TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAA

2. Introduction

3. Introduction

4. Introduction Quality of maps is a property of the collection

5. Introduction Corollary: Individual maps cannot be evaluated in isolation

6. Problem Statement • Input • A collection of related shapes • A collection of maps between all pairs of shapes • A distance measure on each shape

7. Problem Statement • Output: A collection of maps between pairs of shapes that is • Accurate • Consistent

8. Graph Representation

9. Approach • Cycle consistency tells us something about accuracy • Remove the inaccuracies we find • Repeat using the better collection

10. Related Work • Learning Shape Metrics based on Deformations and Transport, Charpiat, NORDIA09 • Dynamic time warping + matching energy for good shortest paths • Disambiguating Visual Relations Using Loop Constraints, Zach et al, CVPR2010 • Use cycle consistency to remove incorrect correspondences • Pairwise mapping methods • Möbius Voting for Surface Correspondence, Lipman et al, SIGGRAPH 2009 • One Point Isometric Matching with the Heat Kernel, Ovsjanikov et al, SGP 2010 • Blended Intrinsic Maps, Kim et al, SIGGRAPH 2011

11. Definitions • Accuracy error: • Consistency error:

12. Relating Cycles to Edges • Call low error “good,” high error “bad” • Good and bad edges cause good and bad cycles • If we can only evaluate the cycles, what can we say about the edges?

13. Relating Cycles to Edges • Accuracy error of a path ° = {i1, …, in} is bounded*: *If ground-truth maps preserve the distortion measure

14. Proposal – Linear Program • For each 3-cycle ° in the graph, compute the distortion C° • Solve the following linear programto find weights for the edges: • Minimize • Subject to • Where

15. Are 3-Cycles Sufficient? A B C D

16. Proposal • LP gives us a weighted graph • Weights give us shortest-path map compositions • But these are just like our input • Run the LP again?

17. Are 3-Cycles Sufficient? A B A B ABD BAC C D C D

18. Proposal - Complete • Repeat the following: • Solve LP => obtain edge-weighted graph • Replace edges with shortest paths • Until one of the following is true: • No edge replacements happen, or • No more 3-cycles are bad

19. Convergence - Experimental Map Type LP Weights Final accuracy

20. Convergence - Theoretical • “Almost-accurate” collection: Each 3-cycle has at most 1 bad map • Every cycle’s distortion is either 0 or equal to the inaccuracy of the 1 bad map • LP weights are exactly the map accuracy errors • Guarantees consistency and accuracy after replacing maps with shortest paths

21. Results – 2D (DTW) Max accuracy error Fraction of maps Max consistency error Fraction of cycles

22. Results – 2D (DTW) Max accuracy error Fraction of maps Max consistency error Fraction of cycles

23. Results – 3D (Möbius Voting)

24. Results – 3D (Heat Kernel) Max consistency error Max accuracy error Fraction of cycles Fraction of maps

25. Results – 3D (Blended Maps) Animals

26. Results – 3D (Blended Maps) Hands

27. Results – 3D (Blended Maps) Humans

28. Results – 3D (Blended Maps) Teddies

29. Future Work • Prove convergence in more general cases • Allow for multiple maps between a given pair of shapes • Discover the structure of the collection using consistency information

30. Conclusions • Collections contain information that allow us to better evaluate maps • Cycle consistency can be used to identify and remove bad maps • Using an LP with 3-cycle constraints lets us do this efficiently • Repeating the process lets us incorporate longer cycles

31. Thank You