1. Numerical geometry of non-rigid shapes
Multidimensional scaling
Alexander Bronstein, Michael Bronstein, Ron Kimmel
© 2007 All rights reserved

2. How to measure shape similarity?
Given two shapes and represented as discrete samples
and , compute their similarity
Extrinsic similarity
Intrinsic similarity

3. Rigid (extrinsic) similarity
Rigid shape similarity: congruence
Degrees of freedom: Euclidean transformations (rotation+translation)
Classical solution: iterative closest point (ICP) algorithm
ROTATION
MATRIX
TRANSLATION VECTOR
Hausdorff distance as a measure of congruence between point clouds
(other distances are usually preferred)
Y. Chen, G. Medioni, 1991
P. J. Besl, N. D. McKay, 1992

4. Cinderella measuring the glass slipper
ICP in fairy tales
Cinderella measuring the glass slipper
Image: Disney

5. Non-rigid (intrinsic) similarity
Congruence is not a good criterion for similarity of non-rigid shapes
Geodesic distances are invariant to isometric deformations and can be easily computed using FMM
Naïve approach: directly compare matrices of geodesic distances
Problem: arbitrary ordering of points (permutation of rows and columns)
degrees of freedom

6. Isometric embedding
Shape
Canonical form
Represent the intrinsic geometry in a Euclidean space by isometrically embedding it into
Treat the resulting images (canonical forms) as rigid shapes, using ICP or other rigid similarity algorithms
Isometric embedding “undoes” the degrees of freedom
A. Elad, R. Kimmel, CVPR, 2001

7. Mapmaking problem A B B A
Earth (sphere)
Planar map

8. Embedding error
Theorema Egregium: a sphere has positive Gaussian curvature, the plane has zero Gaussian curvature, therefore, they are not isometric.
Every cartographer knows: impossible to create a distance-preserving planar map of the Earth!
Does isometric embedding into higher-dimensional spaces exist?

9. Conclusion: generally, isometric embedding does not exist!
Linial’s example A A 1 1 C D 2 A C D 1 B C A 1 1 C D B B 2 C D D 1 1 B
Conclusion: generally, isometric embedding does not exist!

10. Minimum distortion embedding
Find an embedding that distorts the distances the least
Stress function is a measure of distortion
where
Multidimensional scaling (MDS) problem

11. Near-isometric deformations of a shape
Examples of canonical forms
Near-isometric deformations of a shape
Canonical forms
A. Elad, R. Kimmel, CVPR, 2001

12. Matrix expression of the L2-stress
- an matrix of coordinates in the embedding space
- an constant matrix with values
- an matrix-valued function

13. MDS problem variables Non-convex non-linear optimization problem
Using convex optimization techniques is liable to local convergence
Optimum defined up to Euclidean transformation

14. Iterative majorization
Instead of , minimize a convex majorizing function satisfying
Start with some and iteratively update

15. SMACOF algorithm Majorize the stress by a convex quadratic function
Analytic expression for the minimum of :
SMACOF (Scaling by Minimizing a COnvex Function)

16. SMACOF algorithm (cont)
Equivalent to constant-step gradient descent
Guarantees monotonically decreasing sequence of stress values
No guarantee of global convergence
Iteration cost:

17. Application: face recognition
Facial expressions are approximate isometries of the facial surface
Identity = intrinsic geometry
Expression = extrinsic geometry
A. M. Bronstein et al., IJCV, 2005

18. How to canonize a person?
3D surface
acquisition
Cropping
Smoothing
Canonization
A. M. Bronstein et al., IJCV, 2005

19. Application: face recognition
Facial expressions
Canonical forms
A. M. Bronstein et al., IJCV, 2005

20. Application: face recognition
Rigid similarity
Non-rigid similarity
(canonical forms)
A. M. Bronstein et al., IJCV, 2005
Alex
Michael

21. Coarse MDS problem (N=200)
Multiresolution MDS: motivation
Fine MDS problem (N~1000)
Coarse MDS problem (N=200)

22. Multiresolution MDS
Bottom-up: solve coarse grid MDS problem to initialize fine grid problem
Can be performed on multiple resolution levels
Reduce complexity (less fine-grid iterations)
Reduce the risk of local convergence (good initialization)
Solve fine grid problem
Fine grid solution
Fine grid initialization
Interpolate
Solve coarse grid problem
Coarse grid initialization
Coarse grid solution

23. Multigrid MDS Top-down: start with a fine grid initialization
Improve the fine grid initialization by solving a coarse grid problem and propagating the error
Fine grid initialization
Improved solution
Fine grid residual
Decimate
Interpolate
residual
Solve coarse grid problem
Coarse grid initialization
Coarse grid solution
Coarse grid residual

24. Correction
Problem: the minima of fine and coarse grid problems do not coincide!
Add correction term to coarse grid problem to compensate for inconsistency
CORRECTION
Choosing
guarantees that is a coarse grid solution
M. M. Bronstein et al., NLAA, 2006

25. Modified stress
Another problem: the new coarse grid problem is unbounded!
( can be made arbitrarily large by adding a constant to
without changing the stress)
Modified stress: add a quadratic penalty to the stress
thus resolving translation ambiguity (forcing to be centered at the
origin)
M. M. Bronstein et al., NLAA, 2006

26. Plugging everything together
Hierarchy of data
Interpolation and decimation operators to transfer variables and
residuals from one resolution level to another
Hierarchy of optimization problems
Relaxation: optimization algorithm used to improve solution
M. M. Bronstein et al., NLAA, 2006

27. Solve coarsest grid problem
V-cycle
Relax
Decimate
Interpolate and correct
Relax
Relax
Decimate
Interpolate and correct
Solve coarsest grid problem
M. M. Bronstein et al., NLAA, 2006

28. Convergence of SMACOF and MG MDS (N=2145)
Convergence example
Stress
Time (sec)
Convergence of SMACOF and MG MDS (N=2145)
Order of magnitude speedup, especially pronounced for large
M. M. Bronstein et al., NLAA, 2006

29. How to choose the embedding space?
A generic, non-Euclidean embedding space
Must result in small embedding error (good representation)
Convenient representation of points in (local or preferably global
parametrization)
Simple (preferably analytic) expression for distances
The isometry group is simple (few degrees of freedom)

30. Problem: embedding error can be reduced, but not made zero!
Possible choices
Schwartz et al. 1989:
Elad & Kimmel 2001:
Elad & Kimmel 2002:
BBK 2005:
Walter & Ritter 2002:
Euclidean
Spherical
Hyperbolic
Problem: embedding error can be reduced, but not made zero!

31. Generalized MDS Embedding space = triangular mesh Generalized stress
where is the image on triangular mesh
Generalized MDS (GMDS) problem
A. M. Bronstein et al., PNAS, 2006

32. Main differences
Difference 1: the distances have no analytic expression
Consequence 1: geodesic distance interpolation
Difference 2: points represented in local barycentric
coordinates
Consequence 2: optimization with a modified line search (unfolding)
A. M. Bronstein et al., SIAM, 2006

33. Distance interpolation
How to approximate the distances between points ?
Precompute the pair-wise distances between all mesh
vertices using FMM
Find triangles and enclosing
Interpolate from known distances
Interpolate from
A. M. Bronstein et al., SIAM, 2006

34. Modified line search: unfolding
Optimization on triangular mesh requires displacing a point along a ray
(line search)
Line search in barycentric coordinates requires unfolding
Result: polylinear path
A. M. Bronstein et al., SIAM, 2006

35. Conclusions so far
Geodesic distances are intrinsic descriptors of non-rigid shapes invariant to isometric deformations
MDS is an efficient method for representing and comparing intrinsic invariants
Multiresolution and multigrid methods can yield a significant convergence speedup and reduce the risk on local convergence
Generalized MDS allows avoiding the embedding error by embedding one surface into another