Numerical geometry of nonrigid shapes TopologyInvariant Similarity Diffusion

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 1 Topology-Invariant Similarity and Diffusion Geometry Lecture 7 © Alexander & Michael Bronstein tosca. cs. technion. ac. il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 2 Intrinsic similarity – limitations Intrinsically similar Intrinsically dissimilar Suitable for near-isometric Unsuitable for deformations shape deformations modifying shape topology

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 3 Extrinsically similar Intrinsically dissimilar Extrinsically dissimilar Intrinsically dissimilar Desired result: THIS IS THE SAME SHAPE! A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 4 Joint extrinsic/intrinsic similarity DEFORM X TO MATCH Y EXTRINSICALLY CONSTRAIN THE DEFORMATION TO BE AS ISOMETRIC AS POSSIBLE A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 5 Glove fitting example Misfit = Extrinsic dissimilarity Stretching = Intrinsic dissimilarity A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

Extrinsic dissimilarity Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 6 Intrinsic dissimilarity A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 7 Computation of the joint similarity n Optimization variable: the deformed shape vertex coordinates n Assuming has the connectivity of n Split into computation of n Gradients w. r. t. and are required for optimization A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 8 Computation of the extrinsic term n Find and fix correspondence between current n Can be e. g. the closest points n Compute an L 2 variant of a one-sided Hausdorff distance and its gradient n Similar in spirit to ICP A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007 and

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 9 Computation of the intrinsic term n Fix trivial correspondence between and n Compute L 2 distortion of geodesic distances and gradient n is a fixed matrix of all pair-wise geodesic distances on n Can be precomputed using Dijkstra’s algorithm or fast marching A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 10 Computation of the intrinsic term n is function of the optimization variables and needs to be recomputed n First option: modify the Dijkstra’s algorithm or fast marching to compute the gradient in addition to the distance itself n Second option: compute and fix the path of the geodesic n n is a matrix of Euclidean distances between adjacent vertices is a linear operator integrating the path length along fixed path A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 11 Computation of the joint similarity n Alternating minimization algorithm 1 Compute corresponding points 2 Compute shortest paths and assemble 3 Update 4 If change is small, stop; otherwise, go to Step to sufficiently decrease A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007 1

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 12 Numerical example – dataset Data: tosca. cs. technion. ac. il = topology change

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 13 Numerical example – intrinsic similarity no topological changes

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 14 Numerical example – intrinsic similarity Insensitive to strong deformations Sensitive to topological changes = topology-preserving = topology change

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 15 Numerical example – extrinsic similarity Insensitive to topological changes = topology-preserving Sensitive to strong deformations = topology change

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 16 Numerical example – joint similarity Insensitive to topological changes. . . …and to strong deformations = topology-preserving = topology change

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 17 Numerical example – ROC curves 100 False rejection rate (FRR), % Extrinsic EER=10. 3% 10 1 Joint EER=1. 6% Intrinsic EER=7. 7% Intrinsic, no topological changes EER=1. 1% 0. 1 1 10 False acceptance rate (FAR), % 100

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 18 Shape morphing Stronger intrinsic similarity (larger λ) Stronger extrinsic similarity (smaller λ)

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 19 Other intrinsic geometries Geodesic distance is sensitive to topology changes Possible more robust alternatives n “Average path length” n “Density of paths” n Transition probability A. Bronstein, M. Bronstein, R. Kimmel, M. Mahmoudi, G. Sapiro, submitted to IJCV

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 20 Diffusion on manifolds n Kernel (aka affinity function) n Non-negative n Symmetric n Positive semi-definite: for any n Discrete case: symmetric positive semi-definite matrix n Examples: n Adjacency matrix n Heat kernel R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, F. Warner, S. W. Zucker, PNAS 2005

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 21 Diffusion on manifolds n Normalized kernel where n Because of normalization n is no more symmetric n Symmetrized kernel n = probability of step from n Discrete case: Markovian matrix to by random walk (each row sums to 1) R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, F. Warner, S. W. Zucker, PNAS 2005

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 22 Diffusion on manifolds n Diffusion operator n Discrete case: matrix n Spectral theorem: the kernel of operator admits the spectral decomposition where and n Discrete case: are eigenvalues and eigenfunctions of where are eigenvectors of R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, F. Warner, S. W. Zucker, PNAS 2005

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 23 Diffusion on manifolds n Power of the diffusion operator where the kernel is n Discrete case: matrix power n = transition probability from to in m steps R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, F. Warner, S. W. Zucker, PNAS 2005

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 24 Diffusion distance n “Connectivity rate” from to by paths of length m n Small if there are many paths connecting n Large if there are few paths connecting and R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, F. Warner, S. W. Zucker, PNAS 2005

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 25 Diffusion distance n A mathematical exercise: find the kernel of n Discrete case: R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, F. Warner, S. W. Zucker, PNAS 2005

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 26 Diffusion distance Substitute into diffusion distance R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, F. Warner, S. W. Zucker, PNAS 2005

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 27 Diffusion distance n = bump centered at n Becomes wider as m increases n = distance between two bumps n Small if there is “cross-talk” between bumps n Large if bumps do not overlap R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, F. Warner, S. W. Zucker, PNAS 2005

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 28 Kernels

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 29 Diffusion distance Substitute into diffusion distance where R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, F. Warner, S. W. Zucker, PNAS 2005

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 30 Canonical forms, bis n is a metric on n is isometrically embeddable into by means of n Infinitely dimensional canonical form (“diffusion map”) n Truncated gives good convergence rate

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 31 Diffusion maps No topology change Topology change Canonical form Diffusion map

Numerical geometry of non-rigid shapes Topology-Invariant Similarity & Diffusion Geome 32 ® MATLAB intermezzo Diffusion maps