Numerical geometry of nonrigid shapes Isometry invariant similarity

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

Numerical geometry of non-rigid shapes Isometry invariant similarity Invariant similarity SIMILARITY TRANSFORMATION 2

Numerical geometry of non-rigid shapes Isometry invariant similarity Equivalence Equal Congruent Isometric 3

4 Numerical geometry of non-rigid shapes Isometry invariant similarity Equivalence n Equivalence is a binary relation for all on the space of shapes which satisfies n Reflexivity: n Symmetry: n Transitivity: n Can be expressed as a binary function if and only if n Quotient space is the space of equivalence classes

Numerical geometry of non-rigid shapes Isometry invariant similarity Equivalence 5

Numerical geometry of non-rigid shapes Isometry invariant similarity Equivalence All deformations of the human shape are “the same” 6

Numerical geometry of non-rigid shapes Isometry invariant similarity 7 Similarity n Shapes are rarely truly equivalent (e. g. , due to acquisition noise or since most shapes are rigid) n We want to account for “almost equivalence” or similarity n -similar = -isometric (w. r. t. some metric) n Define a distance on the shape space dissimilarity of shapes quantifying the degree of

Numerical geometry of non-rigid shapes Isometry invariant similarity Similarity …than to a human shape A monkey shape is more similar to a deformation of a monkey shape… 8

Numerical geometry of non-rigid shapes Isometry invariant similarity Isometry-invariant distance Non-negative function satisfying for all n Similarity: and are (In particular, are -isometric if and only if ) n Symmetry: n Triangle inequality: Corollary: is a metric on the quotient space Given discretized shapes sampled with radius n Consistency to sampling: -isometric; and 9

Numerical geometry of non-rigid shapes Isometry invariant similarity Canonical forms distance Minimum-distortion embedding Compute Hausdorff distance over all isometries in No fixed embedding space will give distortion-less canonical forms 10

Numerical geometry of non-rigid shapes Isometry invariant similarity Gromov-Hausdorff distance Isometric embedding Gromov-Hausdorff distance: include into minimization Mikhail Gromov 11

Numerical geometry of non-rigid shapes Isometry invariant similarity Properties of Gromov-Hausdorff distance n Metric on the quotient space n Similarity: of isometries of shapes and are -isometric; -isometric n Consistent to sampling: given discretized shapes and sampled with radius n Generalization of Hausdorff distance: n Hausdorff distance between subsets of a metric space n Gromov-Hausdorff distance between metric spaces Gromov, 1981 12

Numerical geometry of non-rigid shapes Isometry invariant similarity Alternative definition I (metric coupling) where n is the disjoint union of n the (semi-) metric Mémoli, 2008 satisfies and 13

Numerical geometry of non-rigid shapes Isometry invariant similarity Alternative definition I (metric coupling) Optimization over A lot of constraints! Mémoli, 2008 translates into finding the values of 14

15 Numerical geometry of non-rigid shapes Isometry invariant similarity Correspondence A subset if for every is called a correspondence between there exists at least one and similarly for every Particular case: given there exists and such that and

Numerical geometry of non-rigid shapes Isometry invariant similarity 16 Correspondence distortion The distortion of correspondence is defined as In the particular case of , consider the following cases for n If the distortion is

Numerical geometry of non-rigid shapes Isometry invariant similarity Correspondence distortion (cont) Case 1 Case 3 Case 2 n Otherwise, the distortion is given by Therefore, 17

Numerical geometry of non-rigid shapes Isometry invariant similarity 18 Alternative definition II (correspondence distortion) Proof sketch 1. Show that for any Since some there exists , by definition of such that Let By triangle inequality, for , with and are subspaces of

Numerical geometry of non-rigid shapes Isometry invariant similarity 19 Alternative definition II (correspondence distortion) 2. Show that for any Let It is sufficient to show that there is a (semi-)metric such that Construct the metric (in particular, , , and as follows for on the disjoint union ).

20 Numerical geometry of non-rigid shapes Isometry invariant similarity Alternative definition II (correspondence distortion) First, For each Since Second, we need to show that On and for is a (semi-)metric on , it is straightforward We only need to show metric properties hold on ,

Numerical geometry of non-rigid shapes Isometry invariant similarity Alternative definition III measures how much is distorted by when embedded into 21

Numerical geometry of non-rigid shapes Isometry invariant similarity Alternative definition III measures how much is distorted by when embedded into 22

Numerical geometry of non-rigid shapes Isometry invariant similarity Alternative definition III measures how far is from being the inverse of 23

Numerical geometry of non-rigid shapes Isometry invariant similarity Generalized MDS A. Bronstein, M. Bronstein & R. Kimmel, 2006 24

Numerical geometry of non-rigid shapes Isometry invariant similarity Discrete Gromov-Hausdorff distance n Two coupled GMDS problems n Can be cast as a constrained problem Bronstein, Bronstein & Kimmel, 2006 25

Numerical geometry of non-rigid shapes Isometry invariant similarity Numerical example Canonical forms (MDS based on 500 points) Bronstein, Bronstein & Kimmel, 2006 Gromov-Hausdorff distance (GMDS based on 50 points) 26

Numerical geometry of non-rigid shapes Isometry invariant similarity 27 Extrinsic similarity using Gromov-Hausdorff distance EXTRINSIC SIMILARITY Congruence ICP distance: Euclidean isometry GH distance with Euclidean metric: Connection between Euclidean GH and ICP distances: Mémoli (2008) Mémoli, 2008

Numerical geometry of non-rigid shapes Isometry invariant similarity Connection to canonical form distance 28

Numerical geometry of non-rigid shapes Isometry invariant similarity Correspondence again A different representation for correspondence using indicator functions defines a valid correspondence if 29

Numerical geometry of non-rigid shapes Isometry invariant similarity 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? 30

Numerical geometry of non-rigid shapes Isometry invariant similarity 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 function Mémoli, 2007 can be considered as a relaxed version of the indicator or as fuzzy correspondence 31

Numerical geometry of non-rigid shapes Isometry invariant similarity 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 32

Numerical geometry of non-rigid shapes Isometry invariant similarity 33 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: n mass transported from n distance traveled Mémoli, 2007 to

34 Numerical geometry of non-rigid shapes Isometry invariant similarity The analogy Hausdorff Wasserstein Distance between subsets of a metric measure of a metric space . Gromov-Hausdorff Distance between metric spaces space . Gromov-Wasserstein Distance between metric measure spaces Mémoli, 2007