Topology-Invariant Similarity and Diffusion Geometry

Transcription

1 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

2 Intrinsic similarity limitations 2 Intrinsically similar Intrinsically dissimilar Suitable for near-isometric shape deformations Unsuitable for deformations modifying shape topology

3 3 Extrinsically dissimilar Intrinsically similar Extrinsically similar Intrinsically dissimilar Extrinsically dissimilar Intrinsically dissimilar Desired result: THIS IS THE SAME SHAPE! A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

4 Joint extrinsic/intrinsic similarity 4 DEFORM X TO MATCH Y EXTRINSICALLY CONSTRAIN THE DEFORMATION TO BE AS ISOMETRIC AS POSSIBLE A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

5 Glove fitting example 5 Misfit = Extrinsic dissimilarity Stretching = Intrinsic dissimilarity A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

6 6 Extrinsic dissimilarity A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007 Intrinsic dissimilarity

7 Computation of the joint similarity 7 Optimization variable: the deformed shape vertex coordinates Assuming has the connectivity of Split into computation of and Gradients w.r.t. are required for optimization A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

8 Computation of the extrinsic term 8 Find and fix correspondence Can be e.g. the closest points between current and Compute an L 2 variant of a one-sided Hausdorff distance and its gradient Similar in spirit to ICP A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

9 Computation of the intrinsic term 9 Fix trivial correspondence between and Compute L 2 distortion of geodesic distances and gradient is a fixed matrix of all pair-wise geodesic distances on Can be precomputed using Dijkstra s algorithm or fast marching A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

10 Computation of the intrinsic term 10 is function of the optimization variables and needs to be recomputed First option: modify the Dijkstra s algorithm or fast marching to compute the gradient in addition to the distance Second option: compute and fix the path of the geodesic itself 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

11 Computation of the joint similarity 11 Alternating minimization algorithm Compute corresponding points Compute shortest paths and assemble Update to sufficiently decrease If change is small, stop; otherwise, go to Step 4 1 A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

12 Numerical example dataset 12 Data: tosca.cs.technion.ac.il = topology change

13 Numerical example intrinsic similarity 13 no topological changes

14 Numerical example intrinsic similarity 14 Insensitive to strong deformations Sensitive to topological changes = topology-preserving = topology change

15 Numerical example extrinsic similarity 15 Insensitive to topological changes Sensitive to strong deformations = topology-preserving = topology change

16 Numerical example joint similarity 16 Insensitive to topological changes... and to strong deformations = topology-preserving = topology change

17 Numerical example ROC curves 17 False rejection rate (FRR), % Joint EER=1.6% Intrinsic, no topological changes EER=1.1% Intrinsic EER=7.7% Extrinsic EER=10.3% False acceptance rate (FAR), %

18 Shape morphing 18 Stronger intrinsic similarity (larger λ) Stronger extrinsic similarity (smaller λ)

19 Other intrinsic geometries 19 Geodesic distance is sensitive to topology changes Possible more robust alternatives Average path length Density of paths Transition probability A. Bronstein, M. Bronstein, R. Kimmel, M. Mahmoudi, G. Sapiro, submitted to IJCV

20 Diffusion on manifolds 20 Kernel (aka affinity function) Non-negative Symmetric Positive semi-definite: for any Discrete case: symmetric positive semi-definite matrix Examples: Adjacency matrix Heat kernel R.R. Coifman, S. Lafon, A.B. Lee, M. Maggioni, F. Warner, S.W. Zucker, PNAS 2005

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

22 Diffusion on manifolds 22 Diffusion operator Discrete case: matrix Spectral theorem: the kernel of operator admits the spectral decomposition where and are eigenvalues and eigenfunctions of Discrete case: where are eigenvectors of R.R. Coifman, S. Lafon, A.B. Lee, M. Maggioni, F. Warner, S.W. Zucker, PNAS 2005

23 Diffusion on manifolds 23 Power of the diffusion operator where the kernel is Discrete case: matrix power = transition probability from to in m steps R.R. Coifman, S. Lafon, A.B. Lee, M. Maggioni, F. Warner, S.W. Zucker, PNAS 2005

24 Diffusion distance 24 Connectivity rate from to by paths of length m Small if there are many paths connecting and 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

25 Diffusion distance 25 A mathematical exercise: find the kernel of Discrete case: R.R. Coifman, S. Lafon, A.B. Lee, M. Maggioni, F. Warner, S.W. Zucker, PNAS 2005

26 Diffusion distance 26 Substitute into diffusion distance R.R. Coifman, S. Lafon, A.B. Lee, M. Maggioni, F. Warner, S.W. Zucker, PNAS 2005

27 Diffusion distance 27 = bump centered at Becomes wider as m increases = distance between two bumps Small if there is cross-talk between bumps Large if bumps do not overlap R.R. Coifman, S. Lafon, A.B. Lee, M. Maggioni, F. Warner, S.W. Zucker, PNAS 2005

28 Kernels 28

29 Diffusion distance 29 Substitute into diffusion distance where R.R. Coifman, S. Lafon, A.B. Lee, M. Maggioni, F. Warner, S.W. Zucker, PNAS 2005

30 Canonical forms, bis 30 is a metric on is isometrically embeddable into by means of Infinitely dimensional canonical form ( diffusion map ) Truncated gives good convergence rate

31 Diffusion maps 31 No topology change Topology change Canonical form Diffusion map