1Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Isometry-Invariant Similarity

It is incredible what Gromov can do just with the triangle

inequality!

D. Sullivan, quoted by M. Berger

Alexander Bronstein, Michael Bronstein© 2008 All rights reserved. Web: tosca.cs.technion.ac.il

2Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Equivalence

Two shapes and are equal if they contain exactly the same

points.

We deem two unequal rigid shapes the same if they are congruent.

Two unequal non-rigid shapes are the same if they are isometric.

Congruence and isometry are equivalence relations.

Formally, equivalence is a binary relation on the space of

shapes which for all satisfies

Reflexivity:

Symmetry:

Transitivity:

Equivalence relation partitions into equivalence classes.

Quotient space is the space of equivalence classes.

3Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Similarity

Equivalence can be expressed as a binary function

, if and only if .

Shapes are rarely truly equivalent (e.g., due to acquisition noise).

We want to account for “almost equivalence” or similarity.

-similar = -isometric (in either intrinsic or extrinsic sense).

Define a distance quantifying the degree of dissimilarity of shapes.

4Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Isometry-invariant distance

Non-negative function satisfying for all

Similarity: and are -isometric;

and are -isometric

(In particular, satisfies the isolation property:

if and only if ).

Symmetry:

Triangle inequality:

Corollary: is a metric on the quotient space .

5Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Discrete isometry-invariant distance

In practice, we work with discrete representations of shapes

and that are -coverings.

We require the discrete version to satisfy two additional properties:

Consistency to sampling:

Efficiency: computation complexity of the approximation is

polynomial.

6Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Canonical forms distance

Given two shapes and .

Compute canonical forms

Compare extrinsic geometries of canonical forms

No fixed embedding space will give distortionless canonical forms.

7Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Gromov-Hausdorff distance

Include into minimization problem

can be selected as disjoint union

equipped with metrics .

and are isometric embeddings.

Alternative:

Felix Hausdorff

Mikhail Gromov

8Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Gromov-Hausdorff distance

A metric on the quotient space of isometries of shapes.

Similarity: and are -isometric;

and are -isometric

Generalization of Hausdorff distance:

Hausdorff distance – distance between subsets of a metric space

Gromov-Hausdorff distance – distance between metric spaces

9Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Gromov-Hausdorff distance

Gromov-Hausdorff distance is computationally intractable!

Fortunately, an alternative formulation exists:

in terms of distortion of embedding of one shape into the other.

Distortion terms

Joint distortion:

10Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Distortion

How much is distorted by when embedded into .

11Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Distortion

How much is distorted by when embedded into .

12Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Joint distortion

How much is far from being the inverse of .

13Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Discrete Gromov-Hausdorff distance

Two coupled GMDS problems

Can be cast as a constrained problem

14Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Discrete Gromov-Hausdorff distance

CANONICAL FORMS (MDS, 500 points)

MINIMUM DISTORTION EMBEDDING(GMDS, 50 points)

15Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Connection to ICP distance

Consider the metric space and rigid shapes and .

Similarity = congruence.

ICP distance:

Gromov-Hausdorff distance:

What is the relation between ICP and Gromov-Hausdorff distances?

16Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Connection to ICP distance

Obviously

Is the converse true?

Theorem [Mémoli, 2008]:

The metrics and are not equal.

Yet, they are equivalent (comparable).

17Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Connection to canonical form distance

18Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Self-similarity (symmetry)

Shape is symmetric, if there exists

a rigid motion

such that .

Am I symmetric?Yes, I am symmetric.

19Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

What about us?

Symmetry

I am symmetric.

20Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Symmetry

Shape is symmetric, if there exists a rigid motion

such that .

Alternatively:

Shape is symmetric if there exists an automorphism

such that .

Said differently:

Shape is symmetric if has a non-trivial self-isometry.

Substitute extrinsic metric with intrinsic counterpart .

Distinguish between extrinsic and intrinsic symmetry.

21Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Symmetry: extrinsic vs. intrinsic

Extrinsic symmetry Intrinsic symmetry

22Numerical geometry of non-rigid shapes Isometry-Invariant Similarity

Symmetry: extrinsic vs. intrinsic

I am extrinsically symmetric. We are extrinsically asymmetric.We are all intrinsically symmetric.