1 numerical geometry of non-rigid shapes non-rigid similarity numerical geometry of non-rigid shapes...

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


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)


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


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


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


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



If , then there exist and such that

.

bijectivity

distance preservation



Given two shapes measure how far they are from being isometric

.

bijectivity





.

bijectivity




Equivalent definition of Gromov-Hausdorff distance in terms of metric

distortions (for compact surfaces):

where:


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)


Computing the Gromov-Hausdorff distance

Generalized MDS problem

Continuous optimization

Deterministic approximation (exact up to numerical accuracy / local

convergence)

BBK (2006)



Gromov-Hausdorff distance via GMDS

Sampling: ,

Optimization over images and

Two coupled GMDS problems



Gromov-Hausdorff distance via GMDS (cont)


Equivalent formulation as a constrained problem using an artificial

variable


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


Example: 3D objects

BBK, SIAM J. Sci. Comp, 2006


Canonical forms distance(MDS, 500 points)

Gromov-Hausdorff distance(GMDS, 50 points)

BBK, SIAM J. Sci. Comp, 2006

Example: 3D objects


Example: Jacobs et al.

Partial similarity

How to compare a centaur to a horse?


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


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

?


Recognition by parts

Divide the shapes into meaningful parts and

Compare each part separately using full similarity criterion

Merge the partial similarities


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?


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


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


Multicriterion optimization

UTOPIA

Minimize the vector objective function over

Competing criteria – impossible to minimize and

simultaneously

ATTAINABLE CRITERIA



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


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,



Scalar- versus set-valued distances

Large Gromov-Hausdorff distanceSmall partial dissimilarity

Large Gromov-Hausdorff distanceLarge partial dissimilarity



Fuzzy approximation


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,


Fuzzy approximation

Crisp part Fuzzy part



Discrete membership functions

Discrete measures

Fuzzy partiality

Fuzzy Gromov-Hausdorff distance


Fuzzy approximation


Alternating minimization

Fix , optimize over

Fix , optimize over

Alternating minimization over

and


Example: mythological creatures

A. M. Bronstein et al., IJCV


Gromov-Hausdorff distance Partial dissimilarity


Example: mythological creatures


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

1 numerical geometry of non-rigid shapes non-rigid similarity numerical geometry of non-rigid shapes...

Documents

rigid objects

isometric embeddings

quotient space slide

mds canonical forms

embedding space

given common metric

metric axioms

terms of metric distortions