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

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

3. 3 Michael Bronstein Computational metric geometry information retrieval shape analysis object detection inverse problemsmedical imaging Similarity

4. 4 Michael Bronstein Computational metric geometry Non-rigid world from macro to nano Animals Organs Micro- organisms Proteins Nano- machines

5. 5 Michael Bronstein Computational metric geometry Rock Paper Scissors Rock, paper, scissors

6. 6 Michael Bronstein Computational metric geometry Hands Rock Paper Scissors Rock, paper, scissors

7. 7 Michael Bronstein Analysis of non-rigid shapes Invariant similarity Similarity Transformation

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

9. 9 Michael Bronstein Computational metric geometry 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. 10 Michael Bronstein Computational metric geometry Examples of metrics GeodesicEuclidean Diffusion

11. 11 Michael Bronstein Computational metric geometry Rigid similarity CongruenceIsometry between metric spaces Min Hausdorff distance over Euclidean isometries Unknown correspondence!

12. 12 Michael Bronstein Computational metric geometry Non-rigid similarity Rigid similarity Part of same metric spaceDifferent metric spaces SOLUTION: Find a representation of and in a common metric space

13. 13 Michael Bronstein Computational metric geometry Canonical forms Elad, Kimmel 2003 Non-rigid shape similarity = Rigid similarity of canonical forms Compute canonical forms Compare canonical forms as rigid shapes

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

15. 15 Michael Bronstein Computational metric geometry Best possible embedding with minimum distortion Multidimensional scaling Non-linear non-convex optimization problem in variables

16. 16 Michael Bronstein Computational metric geometry Interpolate Multigrid MDS B et al. 2005 Fine grid Decimate Solution Coarse grid Improved solution Relax

17. 17 Michael Bronstein Computational metric geometry Multigrid MDS B et al. 2005, 2006 Complexity (MFLOPs) Stress Execution time (sec) Multigrid MDS Standard MDS

18. 18 Michael Bronstein Computational metric geometry Examples of canonical forms

19. 19 Michael Bronstein Computational metric geometry Embedding distortion limits discriminative power!

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

21. 21 Michael Bronstein Computational metric geometry Metric coupling Disjoint union Isometric embedding ? ? How to choose the metric?

22. 22 Michael Bronstein Computational metric geometry Gromov-Hausdorff distance Gromov 1981 Find the smallest possible metric Distance between metric spaces (how isometric two spaces are) Generalization of the Hausdorff distance Gromov-Hausdorff distance

23. 23 Numerical geometry of non-rigid shapes A journey to non-rigid world Canonical formsGromov-Hausdorff Fixed embedding spaceOptimal data-dependent embedding space Approximate metric (error dependent on the data) Metric on equivalence classes of isometric shapes -isometric Consistent to sampling -isometric for shapes sampled at radius

24. 24 Michael Bronstein Computational metric geometry Gromov-Hausdorff distance Gromov 1981 Optimization over all possible correspondences is NP-hard problem! is a correspondence satisfying for every there exists s.t. Theorem: for compact spaces,

25. 25 Michael Bronstein Computational metric geometry Best possible embedding with minimum distortion Multidimensional scaling

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

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

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

29. 29 Numerical Geometry of Non-Rigid Shapes Expression-invariant face recognition Application to face recognition x x’ y y’ Euclidean metric 

30. 30 Numerical Geometry of Non-Rigid Shapes Expression-invariant face recognition Application to face recognition x x’ y y’ Geodesic metric  Distance distortion distribution

31. 31 Michael Bronstein Computational metric geometry

32. 32 Michael Bronstein Computational metric geometry Eikonal vs heat equation Kimmel & Sethian 1998 Weber, Devir, B 2, Kimmel 2008 Viscosity solution: arrival time (geodesic distance from source) Boundary conditions: Initial conditions: Solution : heat distribution at time t

33. 33 Michael Bronstein Computational metric geometry: a new tool in image sciences Heat equation on manifolds 1D3D

34. 34 Michael Bronstein Computational metric geometry: a new tool in image sciences 1D3D Heat kernel Heat equation on manifolds

35. 35 Michael Bronstein Computational metric geometry: a new tool in image sciences 1D3D Heat kernel “Convolution” Heat equation on manifolds

36. 36 Michael Bronstein Computational metric geometry Diffusion distance Geodesic = minimum-length path Diffusion distance = “average” length path (less sensitive to bottlenecks) “Connectivity rate” from to by paths of length Small if there are many paths Large if there are a few paths Berard, Besson, Gallot, 1994; Coifman et al. PNAS 2005

37. 37 Michael Bronstein Computational metric geometry Invariance: Euclidean metric RigidScaleInelastic Topology Wang, B, Paragios 2010

38. 38 Michael Bronstein Computational metric geometry Invariance: geodesic metric RigidScaleInelastic Topology Wang, B, Paragios 2010

39. 39 Michael Bronstein Computational metric geometry Invariance: diffusion metric RigidScaleInelastic Topology Wang, B, Paragios 2010

40. 40 Michael Bronstein Computational metric geometry

41. 41 Michael Bronstein Computational metric geometry shape analysis object detection inverse problemsmedical imaging Similarity information retrieval

42. 42 Michael Bronstein Computational metric geometry Metric learning Representation space “Similar” “Dissimilar” Data space Metric learning: on training set Sampling of Generalization

43. 43 Michael Bronstein Computational metric geometry Similarity-sensitive hashing Hamming spaceData space Shakhnarovich 2005 B 2, Kimmel 2010; Strecha, B, Fua 2010 0001 1111 0100 0011 0111

44. 44 Michael Bronstein Computational metric geometry Luke vs Vader – Starwars classic Lightsaber Original copyPirated copy Video copy detection

45. 45 Michael Bronstein Computational metric geometry C A A A T T G C C Substitution In/Del C C A A T T G C C C C A A T T A G C C B 2, Kimmel 2010 Mutation Substitution In/Del So, what do you think? Biological DNA“Video DNA”

46. 46 Michael Bronstein Computational metric geometry So, what do you think? T positive negative So, what do you think? Mutation-invariant metric B 2, Kimmel 2010

47. 47 Michael Bronstein Computational metric geometry: a new tool in image sciences Gap Gap continuation Pairwise cost Dynamic programming sequence alignment with gaps to account for In/Del mutations (Smith-WATerman algorithm) Optimal alignment = minimum-cost path Learned mutation-invariant pairwise matching cost Video DNA alignment B 2, Kimmel 2010

48. 48 Michael Bronstein Computational metric geometry B 2, Kimmel 2010

49. 49 Michael Bronstein Computational metric geometry B 2, Kimmel 2010

50. 50 Michael Bronstein Computational metric geometry Object similarity is also a metric space Summary Metric space Gromov-Hausdorff distance + GMDS MDSMetric learning Metric choice=invariance Examples of similarity (metric sampling) 0001 1001 1110 1111 0111

51. 51 Michael Bronstein Computational metric geometry Thank you