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

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.

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.

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 .

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.

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.

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

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

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:

Distortion How much is distorted by when embedded into .

Distortion How much is distorted by when embedded into .

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

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

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

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?

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

Connection to canonical form distance

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

Symmetry I am symmetric. What about us?

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.

Symmetry: extrinsic vs. intrinsic Extrinsic symmetry Intrinsic symmetry

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