1. Non-Euclidean Embedding
Out of nothing I have created a strange new universe.
J. Bolyai, on the creation of
non-Euclidean geometry.
Non-Euclidean Embedding
Alexander Bronstein, Michael Bronstein
© 2008 All rights reserved. Web: tosca.cs.technion.ac.il

2. Non-rigid shape similarity?

3. Why non-Euclidean? Reminder: prototype MDS problem with Lp stress
Minimizer is the canonical form of the shape
Minimum is the embedding distortion (in the Lp sense).
Practically, distortion is non-zero.
Sets a data-dependent limit to the discriminative power of the canonical
form-based shape similarity.
Can the distortion be reduced?
Yes, by finding a better embedding space.

4. Non-Euclidean MDS Prototype non-Euclidean MDS problem with Lp stress
Desired properties of :
Convenient representation of points
Preferably global system of coordinates.
Efficient computation of the metric
Preferably closed-form expression.
Simple isometry group
Practical algorithm for “rigid” shape matching in

5. Spherical geometry
-dimensional unit sphere: geometric location of the unit vectors
Sphere of radius obtained by scaling by .
circle, parametrized by
sphere, parametrized by

6. Spherical geometry parametrized by where corresponding to parameters .
Parametrization domain:

7. Spherical geometry Minimal geodesics on the sphere are great circles.
Section of the sphere with the plane passing through origin.
Geodesic distance: length of the arc
Geodesic distance on sphere
of radius
Great circle

8. Spherical embedding Spherical MDS problem:
Embedding into a sphere of radius :
In the limit , we obtain Euclidean embedding into

9. Spherical embedding
Richer geometry than Euclidean (asymptotically Euclidean).
Minimum embedding distortion obtained for shape-dependent radius.

10. What we found?
Global system of coordinates for representing points on the sphere.
Closed-form expression for the metric
Simple isometry group
Orthogonal group.
Origin-preserving rotations in
Smaller embedding distortion.
Complexity similar to Euclidean MDS.

11. What we are still missing?
Embedding distortion still non-zero and depends on data.
No straightforward (ICP-like) algorithm for comparing canonical forms.
Way out:
Embed directly into !

12. Generalized multidimensional scaling
Generalized multidimensional scaling (GMDS)
Multidimensional scaling (MDS)

13. GMDS: croce e delizia Delizia:
Embedding distortion is no more our enemy
Before, it limited the sensitivity of our method.
Now, it quantifies intrinsic dissimilarity of and
Measures how much needs to be changed to fit into
If and are isometric, embedding is distortionless.
No more need to compare canonical forms
Dissimilarity is obtained directly from the embedding distortion.

14. GMDS: croce e delizia Croce: How to represent points on ?
Global parametrization is not always available.
Some local representation is required in general case.
No more closed-form expression for
Metric needs to be approximated.
Minimization algorithm.

15. Local representation is sampled at and represented as a
triangular mesh
Any point falls into one of the triangles
Within the triangle, it can be represented as convex combination
of triangle vertices ,
Barycentric coordinates
We will need to handle discrete indices in minimization algorithm.

16. Geodesic distances Distance terms can be precomputed, since are fixed.
How to compute distance terms ?
No more closed-form expression.
Cannot be precomputed, since are minimization variables.
can fall anywhere on the mesh.
Precompute for all
Approximate
for any

17. Geodesic distance approximation
Approximation from
First order accurate:
Consistent with data:
Symmetric:
Smoothness: is and a closed-form expression
for its derivatives is available to minimization algorithm.
Might be only at some points or along some lines.
Efficiently computed: constant complexity independent of

18. Geodesic distance approximation
Compute for
falls into triangle and is represented as
Particular case:
Hence, we can precompute distances
How to compute from ?

19. Geodesic distance approximation
We have already encountered this problem in fast marching.
Wavefront arrives at triangle vertex at time
When does it arrive to ?
Adopt planar wavefront model.
Distance map is linear in the triangle (hence, linear in )
Solve for coefficients and obtain a linear interpolant

20. Geodesic distance approximation
General case: falls into triangle and is represented as
Apply previous steps in triangle to obtain
Apply once again in triangle to obtain

21. Un ballo a quattro passi

22. Minimization algorithm
How to minimize the generalized stress?
Particular case: L2 stress
Fix all and all except for some
Stress as a function of only becomes quadratic

23. Quadratic stress Quadratic stress is positive semi-definite.
is convex in (but not necessarily in together).

24. Quadratic stress Closed-form solution for minimizer of
Problem: solution might be outside the triangle.
Solution: find constrained minimizer
Closed-form solution still exists.

25. Minimization algorithm
Initialize
For each
Fix and compute gradient
Select corresponding to maximum
Compute minimizer
If constraints are active
translate to adjacent triangle.
Iterate until convergence…

26. How to move to adjacent triangles?
inside
on edge
on vertex
Three cases
All : inside triangle.
: on edge opposite to
: on vertex

27. Point on edge on edge opposite to .
If edge is not shared by any other triangle
we are on the boundary – no translation.
Otherwise, express the point as in triangle .
contains same values as .
May be permuted due to different vertex
ordering in .
Complication: is not on the edge.
Evaluate gradient in
If points inside triangle, update to

28. Point on vertex on vertex . For each triangle sharing vertex
Express point as in
Evaluate gradient in
Reject triangles with pointing
outside.
Select triangle with maximum
Update to

29. MDS vs GMDS MDS Generalized MDS Stress Generalized stress
Analytic expression for
Nonconvex problem
Variables: Euclidean coordinates
of the points
must be interpolated
Nonconvex problem
Variables: points on in
barycentric coordinates

30. Multiresolution Stress is non convex – many small local minima.
Straightforward minimization gives poor results.
How to initialize GMDS?
Multiresolution:
Create a hierarchy of grids in ,
Each grid comprises
Sampling:
Geodesic distance matrix:

31. Multiresolution Initialize at the coarsest resolution in . For
Starting at initialization , solve the GMDS problem
Interpolate solution to next resolution level
Return

32. GMDS
Interpolation
GMDS

33. Multiresolution encore
So far, we created a hierarchy of embedded spaces
One step further: create a hierarchy of embedding spaces

34. MATLAB® intermezzo
GMDS

35. Summary and suggested reading
Spherical embedding
T. F. Cox and M. A. A. Cox, Multidimensional scaling on a sphere,
Communications in Statistics: Theory and Methods 20 (1991), 2943–2953.
T. F. Cox and M. A. A. Cox, Multidimensional scaling, Chapman and Hall, 1994.
Generalized multidimensional scaling
A.M. Bronstein, M.M. Bronstein, and R. Kimmel, Efficient computation of
isometry-invariant distances between surfaces, SIAM J. Scientific Computing 28
(2006), no. 5, 1812–1836.
A.M. Bronstein, M.M. Bronstein, and R. Kimmel, Generalized multidimensional
scaling: a framework for isometry-invariant partial surface matching, PNAS 103
(2006), no. 5, 1168–1172.
Initialization issues
D. Raviv, A. M. Bronstein, M. M. Bronstein, and R. Kimmel, Symmetries of
non-rigid shapes, Proc. NRTL, 2007.