Computational metric geometry
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Michael Bronstein Computational metric geometry Computational metric geometry Michael Bronstein Department of Computer Science Technion – Israel Institute of Technology 1 2 Michael Bronstein Computational metric geometry What is metric geometry? ? Metric space Similarity of metric spaces Metric representation 3 Michael Bronstein Computational metric geometry information retrieval object detection shape analysis inverse problems Similarity medical imaging 4 Michael Bronstein Computational metric geometry Non-rigid world from macro to nano Organs Nanomachines Proteins Microorganisms Animals 5 Michael Bronstein Computational metric geometry Rock, paper, scissors Rock Scissors Paper 6 Michael Bronstein Computational metric geometry Rock, paper, scissors Hands Rock Scissors Paper Michael Bronstein Analysis of non-rigid shapes Invariant similarity Similarity Transformation 7 8 Michael Bronstein Computational metric geometry Metric model Shape Similarity metric space Distance between metric spaces and Invariance isometry w.r.t. . 9 Michael Bronstein Computational metric geometry Isometry Two metric spaces and are isometric if there exists a bijective distance preserving map and map which is distance preserving surjective ‘‘ are -isometric if there exists a ‘‘ Two metric spaces such that -similar = -isometric In which metric? 10 Michael Bronstein Computational metric geometry Examples of metrics Euclidean Geodesic Diffusion 11 Michael Bronstein Computational metric geometry Rigid similarity Isometry between metric spaces Congruence Unknown correspondence! Min Hausdorff distance over Euclidean isometries 12 Michael Bronstein Computational metric geometry Non-rigid similarity Rigid similarity Non-rigid similarity Part of same metric space Different metric spaces SOLUTION: Find a representation of in a common metric space and Michael Bronstein Computational metric geometry Canonical forms Compare canonical forms as rigid shapes Compute canonical forms Non-rigid shape similarity = Rigid similarity of canonical forms Elad, Kimmel 2003 13 Michael Bronstein Computational metric geometry Multidimensional scaling SF 7200 4000 1630 Paris NY TA Rio Find a configuration of points in the plane best representing distances between the cities 14 15 Michael Bronstein Computational metric geometry Multidimensional scaling Best possible embedding with minimum distortion Non-linear non-convex optimization problem in variables 16 Michael Bronstein Computational metric geometry Multigrid MDS Fine grid Solution Decimate Interpolate Relax Coarse grid B et al. 2005 Improved solution 17 Michael Bronstein Computational metric geometry Multigrid MDS Execution time (sec) Multigrid MDS Stress Standard MDS Complexity (MFLOPs) B et al. 2005, 2006 Michael Bronstein Computational metric geometry Examples of canonical forms 18 Michael Bronstein Computational metric geometry Embedding distortion limits discriminative power! 19 20 Michael Bronstein Computational metric geometry Canonical forms, revisited Min distortion embedding Min distortion embedding Fix some metric space Compute Computecanonical Hausdorff forms distance (defined between up tocanonical an isometry forms in ) No fixed (data-independent) embedding space will give distortion-less canonical forms! 21 Michael Bronstein Computational metric geometry Metric coupling Isometric embedding Isometric embedding Disjoint union ? ? How to choose the metric? Michael Bronstein Computational metric geometry 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 Gromov 1981 22 23 Numerical geometry of non-rigid shapes A journey to non-rigid world 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 Michael Bronstein Computational metric geometry Gromov-Hausdorff distance Theorem: for compact spaces, is a correspondence satisfying for every there exists s.t. for every there exists s.t. Optimization over all possible correspondences is NP-hard problem! Gromov 1981 Michael Bronstein Computational metric geometry Multidimensional scaling Best possible embedding with minimum distortion 25 Michael Bronstein Computational metric geometry 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 Michael Bronstein Computational metric geometry 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 27 28 Michael Bronstein Computational metric geometry Canonical forms (MDS based on 500 points) BBK, SIAM J. Sci. Comp 2006 Gromov-Hausdorff distance (GMDS based on 50 points) Numerical Geometry of Non-Rigid Shapes Expression-invariant face recognition Application to face recognition x x’ Euclidean metric y y’ 29 Numerical Geometry of Non-Rigid Shapes Expression-invariant face recognition Application to face recognition x x’ Distance distortion distribution Geodesic metric y y’ 30 Michael Bronstein Computational metric geometry 31 32 Michael Bronstein Computational metric geometry Eikonal vs heat equation Boundary conditions: Viscosity solution: arrival time (geodesic distance from source) Kimmel & Sethian 1998 Weber, Devir, B2, Kimmel 2008 Initial conditions: Solution : heat distribution at time t Michael Bronstein Computational metric geometry: a new tool in image sciences Heat equation on manifolds 1D 3D 33 Michael Bronstein Computational metric geometry: a new tool in image sciences Heat equation on manifolds 1D 3D Heat kernel 34 Michael Bronstein Computational metric geometry: a new tool in image sciences Heat equation on manifolds 1D 3D Heat kernel “Convolution” 35 Michael Bronstein Computational metric geometry 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 37 Michael Bronstein Computational metric geometry Invariance: Euclidean metric Rigid Wang, B, Paragios 2010 Scale Inelastic Topology 38 Michael Bronstein Computational metric geometry Invariance: geodesic metric Rigid Wang, B, Paragios 2010 Scale Inelastic Topology 39 Michael Bronstein Computational metric geometry Invariance: diffusion metric Rigid Wang, B, Paragios 2010 Scale Inelastic Topology Michael Bronstein Computational metric geometry 40 41 Michael Bronstein Computational metric geometry information retrieval object detection shape analysis inverse problems Similarity medical imaging 42 Michael Bronstein Computational metric geometry Metric learning “Similar” “Dissimilar” Generalization Data space Sampling of Metric learning: Representation space on training set 43 Michael Bronstein Computational metric geometry Similarity-sensitive hashing 0001 0011 0100 0111 1111 Data space Shakhnarovich 2005 B2, Kimmel 2010; Strecha, B, Fua 2010 Hamming space 44 Michael Bronstein Computational metric geometry Video copy detection Lightsaber Luke vs Vader – Starwars classic Original copy Pirated copy 45 Michael Bronstein Computational metric geometry Mutation Biological DNA “Video DNA” So, what do you think? C A A T T A G C C A G C C Substitution B2, Kimmel 2010 In/Del Substitution In/Del 46 Michael Bronstein Computational metric geometry Mutation-invariant metric T So, what do you think? positive So, what do you think? So, what do you think? negative So, what do you think? B2, Kimmel 2010 Michael Bronstein Computational metric geometry: a new tool in image sciences Video DNA alignment Gap Pairwise cost Optimal alignment = minimum-cost Gap continuation path 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 47 Michael Bronstein Computational metric geometry B2, Kimmel 2010 48 Michael Bronstein Computational metric geometry B2, Kimmel 2010 49 50 Michael Bronstein Computational metric geometry Summary 0001 1001 1111 0111 1110 MDS Metric space Gromov-Hausdorff Object similarity is also distance a metric + GMDS space Metric choice=invariance Examples of similarity (metric sampling) Metric learning Michael Bronstein Computational metric geometry Thank you 51
