1. Numerical geometry of non-rigid shapes
Non-rigid similarity
Alexander Bronstein, Michael Bronstein, Ron Kimmel
© 2007 All rights reserved. Web: tosca.technion.ac.il

2. Visualization of shape space
Abstract space of deformable shapes (point = shape)
A distance measuring intrinsic similarity of shapes
Equivalence relation: if they are isometric
Similar (small d)
Dissimilar
(large d)
Visualization of shape space

3. Intrinsic similarity properties
Non-negativity:
Symmetry:
Triangle inequality:
Similarity: if then and are isometric
if and are -isometric, then
iff
is a metric on the quotient space
Consistency to sampling: if is a finite -covering of , then
Efficiency: can be efficiently approximated numerically
A. M. Bronstein et al., PNAS, 2006

4. Canonical forms distance
Embed and into a given common metric space by
minimum-distortion embeddings and .
Compare the canonical forms as rigid objects
A. Elad, R. Kimmel, CVPR 2001

5. Canonical forms distance
Canonical form is an approximate representation of intrinsic geometry
(unavoidable embedding error)
satisfies the metric axioms only approximately
Approximately consistent to sampling
Efficient computation using MDS
A. Elad, R. Kimmel, CVPR 2001

6. Gromov-Hausdorff distance
Include the embedding space into the optimization problem
where and are isometric embeddings
Satisfies the metric axioms with
Consistent to sampling: if is an -covering of , then
Computationally intractable
M. Gromov, 1981

7. Gromov-Hausdorff distance
If , then there exist and such that .
distance preservation
bijectivity

8. Gromov-Hausdorff distance
Given two shapes measure how far they are from being isometric .
distance preservation
bijectivity

9. Gromov-Hausdorff distance
Given two shapes measure how far they are from being isometric .
distance preservation
bijectivity

10. Gromov-Hausdorff distance
Given two shapes measure how far they are from being isometric .
distance preservation
bijectivity

11. Gromov-Hausdorff distance
Equivalent definition of Gromov-Hausdorff distance in terms of metric distortions (for compact surfaces):
where:

12. Computing the Gromov-Hausdorff distance
Mémoli & Sapiro (2005)
Replace with a simpler expression
Probabilistic bound on the error
Combinatorial problem
F. Mémoli, G. Sapiro, Foundations Comp. Math, 2005

13. Computing the Gromov-Hausdorff distance
BBK (2006)
Generalized MDS problem
Continuous optimization
Deterministic approximation (exact up to numerical accuracy / local
convergence)
A. M. Bronstein et al., PNAS, 2007

14. Gromov-Hausdorff distance via GMDS
Sampling: ,
Optimization over images and
Two coupled GMDS problems
A. M. Bronstein et al., PNAS, 2007

15. Gromov-Hausdorff distance via GMDS (cont)
Equivalent formulation as a constrained problem using an artificial
variable
A. M. Bronstein et al., PNAS, 2007

16. Gromov-Hausdorff vs. canonical forms
Two stages: embedding and
comparison
Embedding error is a problem
degrading accuracy
Many points (~1000) are
required for accurate comparison
Computational core: MDS
One stage: generalized
embedding
Embedding error is the
measure of similarity
Few points (~10) are required
to compute accurate distortion
Computational core: GMDS

17. Example: 3D objects
BBK, SIAM J. Sci. Comp, 2006

18. Canonical forms distance Gromov-Hausdorff distance
Example: 3D objects
Canonical forms distance
(MDS, 500 points)
Gromov-Hausdorff distance
(GMDS, 50 points)
BBK, SIAM J. Sci. Comp, 2006

19. How to compare a centaur to a horse?
Partial similarity
How to compare a centaur to a horse?
Example: Jacobs et al.

20. Horse is not similar to man
Partial similarity
Horse is similar
to centaur
Man is similar
to centaur
Horse is not similar to man
Partial similarity is an intransitive relation
Non-metric (no triangle inequality)
Weaker than full similarity (shapes may be partially but not fully similar)

21. Human vision example
Recognition of objects according to partial information
Certain parts have more importance in recognition
A significant part is usually sufficient to recognize the entire object ?

22. Recognition by parts Divide the shapes into meaningful parts and
Compare each part separately using full similarity criterion
Merge the partial similarities

23. What are the parts of a shoe?
What is a part?
Problem: how to divide the shapes into parts?
What are the parts of a shoe?
Solution: consider all parts
Optimize over the sets and of all the possible parts of shapes
and :
Technically, and are -algebras

24. Partiality Problem: are all parts equally important?
Just having common parts is insufficient, parts must be significant
Solution: define partiality measuring how large the
selected parts are w.r.t. entire shapes (larger parts = smaller partiality)
Illustration: Herluf Bidstrup

25. Goal: find the largest most similar common part
Partial similarity recipe
Secret sauce ingredients
Sets of all parts
Full similarity criterion (e.g. Gromov-Hausdorff distance)
Partiality e.g.
where are the measures of area
Goal: find the largest most similar common part
A. M. Bronstein et al., SSVM, 2007

26. Multicriterion optimization
Minimize the vector objective function over
Competing criteria – impossible to minimize and simultaneously
ATTAINABLE CRITERIA
UTOPIA
A. M. Bronstein et al., SSVM, 2007

27. Minimum of scalar function
Scalar versus vector optimality
Minimum of scalar function
Pareto optimum
Pareto optimum: a point at which no criterion can be improved without compromising the other
V. Pareto, 1901

28. Pareto distance
Pareto distance: set of all Pareto optima (Pareto frontier), acting as a
set-valued criterion of partial dissimilarity
Only partial order relation exists between set-valued distances: not
always possible to compare
Infinite possibilities to convert Pareto distance into a scalar-valued one
One possibility: select a point on the
Pareto frontier closest to the utopia
point,
A. M. Bronstein et al., SSVM, 2007

29. Scalar- versus set-valued distances
Large Gromov-Hausdorff distance
Small partial dissimilarity
Large Gromov-Hausdorff distance
Large partial dissimilarity
A. M. Bronstein et al., SSVM, 2007

30. Fuzzy approximation
Problem: Optimization over subsets is an NP-hard problem
( possible parts)
Solution: fuzzy approximation
A part can be represented by the binary function
Relax the problem: define membership function, which can obtain continuous values,
A. M. Bronstein et al., SSVM, 2007

31. Fuzzy approximation Crisp part Fuzzy part
A. M. Bronstein et al., SSVM, 2007

32. Fuzzy approximation Discrete membership functions Discrete measures
Fuzzy partiality
Fuzzy Gromov-Hausdorff distance
A. M. Bronstein et al., SSVM, 2007

33. Alternating minimization
Alternating minimization over and
Fix , optimize over
Fix , optimize over

34. Example: mythological creatures
A. M. Bronstein et al., IJCV

35. Gromov-Hausdorff distance Partial dissimilarity
Example: mythological creatures
Gromov-Hausdorff distance
Partial dissimilarity
A. M. Bronstein et al., SSVM, 2007

36. Conclusions so far
Axiomatic construction of isometry-invariant distances on the space of non-rigid shapes
Gromov-Hausdorff computation using GMDS
Pareto formalism for partial similarity of shapes