1. 1 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Diffusion geometry © Alexander & Michael Bronstein, 2006-2009 © Michael Bronstein, 2010 tosca.cs.technion.ac.il/book 048921 Advanced topics in vision Processing and Analysis of Geometric Shapes EE Technion, Spring 2010

2. 2 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Diffusion geometry 1. Beylkin & Niyogi 2003; Coifman, Lafon, Lee, Maggioni, Warner & Zucker 2005 2. Florack & ter Haar Romeny 1992 3. Sochen, Kimmel, Bruckstein 2001; Dascal, Ditkowski & Sochen 2009 Image processing Edge-preserving filtering, anisotropic diffusion 3 Data analysis Dimensionality reduction, manifold learning 1 Computer vision Scale-space feature detection 2

3. 3 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Diffusion kernel Given manifold with a measure, a diffusion kernel is a function satisfying Non-negativity Symmetry Positive semi-definiteness Square integrability Conservation Can be interpreted as transition probability from to

4. 4 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Diffusion kernel In the discrete setting, a diffusion kernel is an N  N matrix satisfying Non-negativity Symmetry Positive semi-definiteness Conservation (stochastic matrix)

5. 5 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Diffusion operator Diffusion kernel defines a diffusion operator for every function on Can be interpreted as a generalized (non-shift-invariant) convolution of with a space-dependent impulse response

6. 6 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Spectral properties Positive semidefiniteness: for every admits discrete orthogonal eigendecomposition Spectral theorem:

7. 7 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Spectral properties admits discrete orthogonal eigendecomposition Spectral theorem: Positive semidefiniteness: Non-negativity+Conservation: by Perron-Frobenius theorem

8. 8 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Spectral properties Square integrability: (Parseval’s theorem) admits discrete orthogonal eigendecomposition Spectral theorem:

9. 9 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Spectral properties A diffusion operator admits orthogonal eigendecomposition with square-summable eigenvalues The associated diffusion kernel is given by For any, is also a diffusion operator with eigendecomposition and kernel Can be interpreted as transition probability by random walk of steps

10. 10 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Diffusion scale space Important particular case: diffusion operators diagonalized by eignebasis of Laplace-Beltrami operator Operator can be expressed as transfer function = frequency = low-pass filter can be interpreted as multiple filter applications (application of ) Creates a scale-space with scale parameter Small : fast decay of in space, slow decay of in freq. Large : slow decay of in space, fast decay of in freq.

11. 11 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Heat diffusion on manifolds Initial conditions: heat distribution at time Boundary conditions (if X has a boundary) Solution : heat distribution at time Heat equation governs heat propagation on a manifold X

12. 12 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Heat kernel is a fundamental solution of the heat equation with point heat source at (heat value at point after time ) Heat kernel

13. 13 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Heat operator Coifman, Lafon, Lee, Maggioni, Warner & Zucker 2005 Kernel can be represented in the Laplace-Beltrami eigenbasis as Heat kernel is the diffusion kernel of the heat operator Coefficients decay fast: approximation by truncated sum

14. 14 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Probabilistic interpretation Brownian motion on X starting at point x, measurable set C probability of the Brownian motion to be in C at time t Coifman, Lafon, Lee, Maggioni, Warner & Zucker 2005 Heat kernel represents transition probability

15. 15 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Geometric properties In the plane, heat kernel is Gaussian On non-Euclidean shapes

16. 16 Numerical Geometry of Non-Rigid Shapes Diffusion 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 Scale parameter Berard, Besson, Gallot, 1994; Coifman et al. PNAS 2005

17. 17 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Diffusion maps Berard, Besson, Gallot, 1994; Coifman et al. PNAS 2005

18. 18 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Diffusion maps Berard, Besson, Gallot, 1994; Coifman et al. PNAS 2005 Diffusion map Isometric embedding of diffusion distance Infinitely-dimensional canonical form

19. 19 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Diffusion maps Berard, Besson, Gallot, 1994; Coifman et al. PNAS 2005 Finite-dimensional approximation Ambiguity Simple eigenvalues: reflection Eigenvalues with multiplicity: rotation in corresponding eigenspaces

20. 20 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Why diffusion geometry? Intrinsic (hence, deformation-invariant) Robust Easily computed Computable on any shape representation (point cloud, mesh, etc.) Multi-scale Common denominator of many methods

21. 21 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Rigid Invariance InelasticTopological Scale

22. 22 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Scale invariance Scaled by Original shape

23. 23 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Scale invariance Eigenfunctions Scaled by Not scale invariant! Eigenvalues Diffusion map Original shape Heat kernel Diffusion distance

24. 24 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Scale invariance Solution 1: find scale-invariant diffusion operator demanding Corresponding diffusion distance Scale invariant! BB, PAMI 2010

25. 25 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Scale invariance Solution 2: integrate diffusion distance over all scales Results are equivalent up to factor.

26. 26 Numerical Geometry of Non-Rigid Shapes Diffusion Geometry Commute time distance Connectivity of x and y by walks of length t Connectivity of x and y by walks of any length Chandra et al. 1989; Klein & Randic 1993; Saerens, Fouss, Yen, Dupont 2004; Fouss, Pirotte, Renders, Saerens 2007