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 1

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

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

4. CONSTRAIN THE DEFORMATION TO BE AS ISOMETRIC AS POSSIBLE
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

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

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

7. Computation of the joint similarity
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
Find and fix correspondence between current and
Can be e.g. the closest points
Compute an L2 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
Fix trivial correspondence between and
Compute L2 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
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 itself
Second option: compute and fix the path of the geodesic
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
Alternating minimization algorithm
Compute corresponding points
Compute shortest paths and assemble
Update to sufficiently decrease
If change is small, stop; otherwise, go to Step 1 2 3 4 1
A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

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

13. no topological changes
Numerical example – intrinsic similarity
no topological changes

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

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

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

17. Numerical example – ROC curves
100
Extrinsic
EER=10.3%
Joint
Intrinsic 10
EER=1.6%
EER=7.7%
False rejection rate (FRR), % 1
Intrinsic,
no topological
changes
EER=1.1%
0.1
0.1 1 10
100
False acceptance rate (FAR), %

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

19. Other intrinsic geometries
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
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
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
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
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 “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 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 Substitute into diffusion distance
R.R. Coifman, S. Lafon, A.B. Lee, M. Maggioni, F. Warner, S.W. Zucker, PNAS 2005

27. Diffusion distance = 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

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

30. Canonical forms, bis 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 No topology change Topology change Canonical form

32. MATLAB® intermezzo
Diffusion maps 32