Isometry invariant similarity Lecture 7 © Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009 1

Invariant similarity SIMILARITY TRANSFORMATION 2

Equivalence Equal Congruent Isometric

Equivalence Equivalence is a binary relation on the space of shapes which for all satisfies Reflexivity: Symmetry: Transitivity: Can be expressed as a binary function if and only if Quotient space is the space of equivalence classes

Equivalence

All deformations of the human shape are “the same” Equivalence All deformations of the human shape are “the same”

Similarity Shapes are rarely truly equivalent (e.g., due to acquisition noise or since most shapes are rigid) We want to account for “almost equivalence” or similarity -similar = -isometric (w.r.t. some metric) Define a distance on the shape space quantifying the degree of dissimilarity of shapes

A monkey shape is more similar to a deformation of a monkey shape… Similarity …than to a human shape A monkey shape is more similar to a deformation of a monkey shape…

Isometry-invariant distance Non-negative function satisfying for all Similarity: and are -isometric; and are -isometric (In particular, if and only if ) Symmetry: Triangle inequality: Corollary: is a metric on the quotient space Given discretized shapes and sampled with radius Consistency to sampling:

Compute Hausdorff distance over all isometries in Canonical forms distance Minimum-distortion embedding Minimum-distortion embedding Compute Hausdorff distance over all isometries in No fixed embedding space will give distortion-less canonical forms

Gromov-Hausdorff distance Isometric embedding Isometric embedding Gromov-Hausdorff distance: include into minimization Mikhail Gromov

Properties of Gromov-Hausdorff distance Metric on the quotient space of isometries of shapes Similarity: and are -isometric; and are -isometric Consistent to sampling: given discretized shapes and sampled with radius Generalization of Hausdorff distance: Hausdorff distance between subsets of a metric space Gromov-Hausdorff distance between metric spaces Gromov, 1981

Alternative definition I (metric coupling) where is the disjoint union of and the (semi-) metric satisfies and Mémoli, 2008

Alternative definition I (metric coupling) Optimization over translates into finding the values of A lot of constraints! Mémoli, 2008

Correspondence A subset is called a correspondence between and if for every there exists at least one such that and similarly for every there exists such that Particular case: given and

Correspondence distortion The distortion of correspondence is defined as In the particular case of , consider the following cases for If the distortion is

Correspondence distortion (cont) Case 1 Case 3 Case 2 Otherwise, the distortion is given by Therefore,

Alternative definition II (correspondence distortion) Proof sketch 1. Show that for any there exists with Since , by definition of , and are subspaces of some such that Let By triangle inequality, for

Alternative definition II (correspondence distortion) 2. Show that for any Let It is sufficient to show that there is a (semi-)metric on the disjoint union such that , , and Construct the metric as follows (in particular, for ).

Alternative definition II (correspondence distortion) First, For each Since for , Second, we need to show that is a (semi-)metric on On and , it is straightforward We only need to show metric properties hold on

Alternative definition III measures how much is distorted by when embedded into

Alternative definition III measures how much is distorted by when embedded into

Alternative definition III measures how far is from being the inverse of

Generalized MDS A. Bronstein, M. Bronstein & R. Kimmel, 2006

Discrete Gromov-Hausdorff distance Two coupled GMDS problems Can be cast as a constrained problem Bronstein, Bronstein & Kimmel, 2006

Gromov-Hausdorff distance Numerical example Canonical forms (MDS based on 500 points) Gromov-Hausdorff distance (GMDS based on 50 points) Bronstein, Bronstein & Kimmel, 2006

Extrinsic similarity using Gromov-Hausdorff distance Congruence Euclidean isometry ICP distance: GH distance with Euclidean metric: Connection between Euclidean GH and ICP distances: Mémoli (2008) Mémoli, 2008

Connection to canonical form distance

Correspondence again A different representation for correspondence using indicator functions defines a valid correspondence if

Lp Gromov-Hausdorff distance We can give an alternative formulation of the Gromov-Hausdorff distance Can we define an Lp version of the Gromov-Hausdorff distance by relaxing the above definition?

Measure coupling Let be probability measures defined on and (a metric space with measure is called a metric measure or mm space) A measure on is a coupling of and if for all measurable sets The measure can be considered as a relaxed version of the indicator function or as fuzzy correspondence Mémoli, 2007

Gromov-Wasserstein distance The relaxed version of the Gromov-Hausdorff distance is given by and is referred to as Gromov-Wasserstein distance [Memoli 2007] Mémoli, 2007

Earth Mover’s distance Let be a metric space, and measures supported on Define the coupling of The Wasserstein or Earth Mover’s distance (EMD) is given by EMD as optimal mass transport: mass transported from to distance traveled Mémoli, 2007

The analogy Hausdorff Wasserstein Gromov-Hausdorff Gromov-Wasserstein Distance between subsets of a metric space . Distance between subsets of a metric measure space . Gromov-Hausdorff Gromov-Wasserstein Distance between metric spaces Distance between metric measure spaces Mémoli, 2007