1. Isometry-Invariant Similarity
It is incredible what Gromov can do just with the triangle inequality!
D. Sullivan, quoted by M. Berger
Isometry-Invariant Similarity
Alexander Bronstein, Michael Bronstein
© 2008 All rights reserved. Web: tosca.cs.technion.ac.il

2. 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.

3. 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.

4. 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

5. 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.

6. 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.

7. 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

8. 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

9. 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:

10. Distortion
How much is distorted by when embedded into

11. Distortion
How much is distorted by when embedded into

12. Joint distortion
How much is far from being the inverse of

13. Discrete Gromov-Hausdorff distance
Two coupled GMDS problems
Can be cast as a constrained problem

14. MINIMUM DISTORTION EMBEDDING
Discrete Gromov-Hausdorff distance
CANONICAL FORMS
(MDS, 500 points)
MINIMUM DISTORTION EMBEDDING
(GMDS, 50 points)

15. 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?

16. Connection to ICP distance
Obviously
Is the converse true?
Theorem [Mémoli, 2008]:
The metrics and are not equal.
Yet, they are equivalent (comparable).

17. Connection to canonical form distance

18. Self-similarity (symmetry)
Shape is symmetric, if there exists
a rigid motion
such that
Yes, I am symmetric.
Am I symmetric?

19. Symmetry
I am symmetric.
What about us?

20. 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.

21. Symmetry: extrinsic vs. intrinsic
Extrinsic symmetry
Intrinsic symmetry

22. Symmetry: extrinsic vs. intrinsic
I am extrinsically symmetric.
We are all intrinsically symmetric.
We are extrinsically asymmetric.