1Numerical geometry of non-rigid shapes Non-rigid similarity
Numerical geometry of non-rigid shapes
Non-rigid similarity
Alexander Bronstein, Michael Bronstein, Ron Kimmel© 2007 All rights reserved. Web: tosca.technion.ac.il
2Numerical geometry of non-rigid shapes Non-rigid similarity
Abstract space of deformable shapes (point = shape)
A distance measuring intrinsic similarity of shapes
Equivalence relation: if they are isometric
Shape space
Visualization of shape space
Dissimilar(large d)
Similar (small d)
3Numerical geometry of non-rigid shapes Non-rigid similarity
Intrinsic similarity properties
Non-negativity:
Symmetry:
Triangle inequality:
Similarity: if then and are -isometric
if and are -isometric, then
iff
A. M. Bronstein et al., PNAS, 2006
Consistency to sampling: if is a finite -covering of , then
Efficiency: can be efficiently approximated numerically
is a metric on the quotient space
4Numerical geometry of non-rigid shapes Non-rigid similarity
Canonical forms distance
A. Elad, R. Kimmel, CVPR 2001
Embed and into a given common metric space by
minimum-distortion embeddings and .
Compare the canonical forms as rigid objects
5Numerical geometry of non-rigid shapes Non-rigid similarity
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
Canonical forms distance
A. Elad, R. Kimmel, CVPR 2001
6Numerical geometry of non-rigid shapes Non-rigid similarity
Gromov-Hausdorff distance
M. Gromov, 1981
Include the embedding space into the optimization problem
Satisfies the metric axioms with
Consistent to sampling: if is an -covering of , then
Computationally intractable
where and are isometric embeddings
7Numerical geometry of non-rigid shapes Non-rigid similarity
Gromov-Hausdorff distance
If , then there exist and such that
.
bijectivity
distance preservation
8Numerical geometry of non-rigid shapes Non-rigid similarity
Gromov-Hausdorff distance
Given two shapes measure how far they are from being isometric
.
bijectivity
distance preservation
9Numerical geometry of non-rigid shapes Non-rigid similarity
Gromov-Hausdorff distance
Given two shapes measure how far they are from being isometric
.
bijectivity
distance preservation
10Numerical geometry of non-rigid shapes Non-rigid similarity
Gromov-Hausdorff distance
Given two shapes measure how far they are from being isometric
.
bijectivity
distance preservation
11Numerical geometry of non-rigid shapes Non-rigid similarity
Gromov-Hausdorff distance
Equivalent definition of Gromov-Hausdorff distance in terms of metric
distortions (for compact surfaces):
where:
12Numerical geometry of non-rigid shapes Non-rigid similarity
Computing the Gromov-Hausdorff distance
F. Mémoli, G. Sapiro, Foundations Comp. Math, 2005
Replace with a simpler expression
Probabilistic bound on the error
Combinatorial problem
Mémoli & Sapiro (2005)
13Numerical geometry of non-rigid shapes Non-rigid similarity
Computing the Gromov-Hausdorff distance
Generalized MDS problem
Continuous optimization
Deterministic approximation (exact up to numerical accuracy / local
convergence)
BBK (2006)
A. M. Bronstein et al., PNAS, 2007
14Numerical geometry of non-rigid shapes Non-rigid similarity
Gromov-Hausdorff distance via GMDS
Sampling: ,
Optimization over images and
Two coupled GMDS problems
A. M. Bronstein et al., PNAS, 2007
15Numerical geometry of non-rigid shapes Non-rigid similarity
Gromov-Hausdorff distance via GMDS (cont)
A. M. Bronstein et al., PNAS, 2007
Equivalent formulation as a constrained problem using an artificial
variable
16Numerical geometry of non-rigid shapes Non-rigid similarity
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
CANONICAL FORMS GROMOV-HAUSDORFF
17Numerical geometry of non-rigid shapes Non-rigid similarity
Example: 3D objects
BBK, SIAM J. Sci. Comp, 2006
18Numerical geometry of non-rigid shapes Non-rigid similarity
Canonical forms distance(MDS, 500 points)
Gromov-Hausdorff distance(GMDS, 50 points)
BBK, SIAM J. Sci. Comp, 2006
Example: 3D objects
19Numerical geometry of non-rigid shapes Non-rigid similarity
Example: Jacobs et al.
Partial similarity
How to compare a centaur to a horse?
20Numerical geometry of non-rigid shapes Non-rigid similarity
Partial similarity
Partial similarity is an intransitive relation
Non-metric (no triangle inequality)
Weaker than full similarity (shapes may be partially but not fully similar)
Horse is similar to centaur
Man is similar to centaur
Horse is not similar to man
21Numerical geometry of non-rigid shapes Non-rigid similarity
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
?
22Numerical geometry of non-rigid shapes Non-rigid similarity
Recognition by parts
Divide the shapes into meaningful parts and
Compare each part separately using full similarity criterion
Merge the partial similarities
23Numerical geometry of non-rigid shapes Non-rigid similarity
Solution: consider all parts
Optimize over the sets and of all the possible parts of
shapes
and :
What is a part?
Problem: how to divide the shapes into parts?
Technically, and are -algebras
What are the parts of a shoe?
24Numerical geometry of non-rigid shapes Non-rigid similarity
Problem: are all parts equally important?
Partiality
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
25Numerical geometry of non-rigid shapes Non-rigid similarity
Partial similarity recipe
Secret sauce ingredients
Sets of all parts
Full similarity criterion (e.g. Gromov-Hausdorff distance)
Partiality e.g.
A. M. Bronstein et al., SSVM, 2007
Goal: find the largest most similar common part
where are the measures of area
26Numerical geometry of non-rigid shapes Non-rigid similarity
Multicriterion optimization
UTOPIA
Minimize the vector objective function over
Competing criteria – impossible to minimize and
simultaneously
ATTAINABLE CRITERIA
A. M. Bronstein et al., SSVM, 2007
27Numerical geometry of non-rigid shapes Non-rigid similarity
Scalar versus vector optimality
V. Pareto, 1901
Minimum of scalar function Pareto optimum
Pareto optimum: a point at which no criterion can be improved without
compromising the other
28Numerical geometry of non-rigid shapes Non-rigid similarity
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
29Numerical geometry of non-rigid shapes Non-rigid similarity
Scalar- versus set-valued distances
Large Gromov-Hausdorff distanceSmall partial dissimilarity
Large Gromov-Hausdorff distanceLarge partial dissimilarity
A. M. Bronstein et al., SSVM, 2007
30Numerical geometry of non-rigid shapes Non-rigid similarity
Fuzzy approximation
A. M. Bronstein et al., SSVM, 2007
Solution: fuzzy approximation
A part can be represented by the binary function
Problem: Optimization over subsets is an NP-hard problem
( possible parts)
Relax the problem: define membership function, which can obtain
continuous values,
31Numerical geometry of non-rigid shapes Non-rigid similarity
Fuzzy approximation
Crisp part Fuzzy part
A. M. Bronstein et al., SSVM, 2007
32Numerical geometry of non-rigid shapes Non-rigid similarity
Discrete membership functions
Discrete measures
Fuzzy partiality
Fuzzy Gromov-Hausdorff distance
A. M. Bronstein et al., SSVM, 2007
Fuzzy approximation
33Numerical geometry of non-rigid shapes Non-rigid similarity
Alternating minimization
Fix , optimize over
Fix , optimize over
Alternating minimization over
and
34Numerical geometry of non-rigid shapes Non-rigid similarity
Example: mythological creatures
A. M. Bronstein et al., IJCV
35Numerical geometry of non-rigid shapes Non-rigid similarity
Gromov-Hausdorff distance Partial dissimilarity
A. M. Bronstein et al., SSVM, 2007
Example: mythological creatures
36Numerical geometry of non-rigid shapes Non-rigid similarity
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
Conclusions so far