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Computational metric geometry

Computational metric geometry. Michael Bronstein. Department of Computer Science Technion – Israel Institute of Technology. What is metric geometry?. ?. Metric space. Similarity of metric spaces. Metric representation. information retrieval. shape analysis. inverse problems.

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Computational metric geometry

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  1. Computational metric geometry Michael Bronstein Department of Computer Science Technion – Israel Institute of Technology

  2. What is metric geometry? ? Metric space Similarity of metric spaces Metric representation

  3. information retrieval shape analysis inverse problems medical imaging object detection Similarity

  4. Non-rigid world from macro to nano Organs Proteins Nano- machines Micro- organisms Animals

  5. Rock, paper, scissors Rock Scissors Paper

  6. Rock, paper, scissors Hands Rock Scissors Paper

  7. Invariant similarity Similarity Transformation

  8. Metric model  Shape Similarity Invariance metric space Distance between metric spaces and . isometry w.r.t.

  9. ‘ ‘ ‘ ‘ Isometry Two metric spaces and are isometric if there exists a bijective distance preserving map such that Two metric spaces and are -isometric if there exists a map which is • distance preserving • surjective -isometric -similar = In which metric?

  10. Examples of metrics Euclidean Geodesic Diffusion

  11. Rigid similarity Isometry between metric spaces Congruence Unknown correspondence! Min Hausdorff distance over Euclidean isometries

  12. Non-rigid similarity Rigid similarity Non-rigid similarity Part of same metric space Different metric spaces SOLUTION: Find a representation of and in a common metric space

  13. Canonical forms ? Compare canonical forms as rigid shapes Compute canonical forms Non-rigid shape similarity = Rigid similarity of canonical forms Elad, Kimmel 2003

  14. Multidimensional scaling 7200 4000 5200 1630 TA SF 1800 Paris NY 2200 1900 2350 3100 Rio Find a configuration of points in the plane best representing distances between the cities

  15. Multidimensional scaling Best possible embedding with minimum distortion Non-linear non-convex optimization problem in variables

  16. Multigrid MDS Fine grid Solution Decimate Interpolate Relax Coarse grid Improved solution B et al. 2005

  17. Multigrid MDS Execution time (sec) Multigrid MDS Standard MDS Stress Complexity (MFLOPs) B et al. 2005, 2006

  18. Examples of canonical forms

  19. Embedding distortion limits discriminative power!

  20. Compute canonical forms (defined up to an isometry in ) Canonical forms, revisited Min distortion embedding Min distortion embedding Fix some metric space Compute Hausdorff distance between canonical forms No fixed (data-independent) embedding space will give distortion-less canonical forms!

  21. Metric coupling Isometric embedding Isometric embedding Disjoint union ? ? How to choose the metric?

  22. Gromov-Hausdorff distance Find the smallest possible metric Gromov-Hausdorff distance Distance between metric spaces (how isometric two spaces are) Generalization of the Hausdorff distance Gromov1981

  23. Canonical forms Gromov-Hausdorff Fixed embedding space Optimal data-dependent embedding space Approximate metric (error dependent on the data) Metric on equivalence classes of isometric shapes -isometric -isometric Consistent to sampling for shapes sampled at radius

  24. Gromov-Hausdorff distance Theorem: for compact spaces, is a correspondencesatisfying for every there exists s.t. for every there exists s.t. Optimization over all possible correspondences is NP-hard problem! Gromov1981

  25. Multidimensional scaling Best possible embedding with minimum distortion

  26. Generalized multidimensional scaling Best possible embedding with minimum distortion • Geodesic distances have no closed-form expression • No global representation for optimization variables • How to perform optimization on a manifold? B et al. 2006

  27. GMDS: some technical details • No global system of coordinates Use local barycentric coordinates • No closed-form distances Interpolate distances from those pre-computed on the mesh • How to perform optimization? Perform path unfolding to go across triangles B et al. 2005

  28. Canonical forms (MDS based on 500 points) Gromov-Hausdorff distance (GMDS based on 50 points) BBK, SIAM J. Sci. Comp 2006

  29. Application to face recognition  x y y’ x’ Euclidean metric

  30. Application to face recognition  x y y’ x’ Distance distortion distribution Geodesic metric

  31. Eikonalvs heat equation Boundary conditions: Initial conditions: Viscosity solution: arrival time (geodesic distance from source) Solution : heat distribution at time t Kimmel & Sethian 1998 Weber, Devir, B2, Kimmel 2008

  32. Heat equation on manifolds 1D 3D

  33. Heat equation on manifolds 1D 3D Heat kernel

  34. Heat equation on manifolds 1D 3D Heat kernel “Convolution”

  35. Diffusion distance “Connectivity rate” from to by paths of length • Small if there are many paths • Large if there are a few paths Geodesic = minimum-length path Diffusion distance = “average” length path (less sensitive to bottlenecks) Berard, Besson, Gallot, 1994; Coifman et al. PNAS 2005

  36. Invariance: Euclidean metric Topology Rigid Scale Inelastic Wang, B, Paragios 2010

  37. Invariance: geodesic metric Topology Rigid Scale Inelastic Wang, B, Paragios 2010

  38. Invariance: diffusion metric Topology Rigid Scale Inelastic Wang, B, Paragios 2010

  39. information retrieval shape analysis inverse problems medical imaging object detection Similarity

  40. Metric learning “Similar” “Dissimilar” Generalization Data space Representation space Sampling of Metric learning: on training set

  41. Similarity-sensitive hashing 0001 0011 0100 0111 1111 Data space Hamming space Shakhnarovich 2005 B2, Kimmel 2010; Strecha, B, Fua 2010

  42. Video copy detection Lightsaber Luke vs Vader – Starwars classic Original copy Pirated copy

  43. Mutation Biological DNA “Video DNA” So, what do you think? C C A A T T G C C C A AA T T G C C C C A A T TA G C C Substitution In/Del In/Del Substitution B2, Kimmel 2010

  44. Mutation-invariant metric T So, what do you think? So, what do you think? So, what do you think? positive So, what do you think? So, what do you think? negative B2, Kimmel 2010

  45. Video DNA alignment Gap Pairwise cost Optimal alignment = minimum-cost path Gap continuation Gap • Dynamic programming sequence alignment with gaps to account for In/Del mutations (Smith-WATerman algorithm) • Learned mutation-invariant pairwise matching cost B2, Kimmel 2010

  46. B2, Kimmel 2010

  47. B2, Kimmel 2010

  48. Summary 0001 1001 1111 0111 1110 MDS Metric space Metric learning Object similarity is also a metric space Gromov-Hausdorff distance + GMDS Metric choice=invariance Examples of similarity (metric sampling)

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