Invariant shape similarity. Invariant shape similarity. Invariant similarity. Equivalence. Equivalence. Equivalence. Equal SIMILARITY TRANSFORMATION

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1 1 Invariant shape similarity Alexer & Michael Bronstein, Michael Bronstein, 2010 tosca.cs.technion.ac.il/book 2 Invariant shape similarity Advanced topics in vision Processing Analysis of Geometric Shapes EE Technion, Spring Invariant similarity Alexer & Michael Bronstein, Michael Bronstein, 2010 tosca.cs.technion.ac.il/book Advanced topics in vision Processing Analysis of Geometric Shapes EE Technion, Spring Equivalence SIMILARITY Equal TRANSFORMATION Congruent 5 Equivalence Equivalence Equivalence is a binary relation for all on the space of shapes which satisfies Reflexivity: Symmetry: Transitivity: Can be expressed as a binary function if only if Quotient space 1 is the space of equivalence classes Isometric 6

2 7 Equivalence 8 Similarity Shapes are rarely truly equivalent (e.g., due to acquisition noise or since All deformations of the most shapes are rigid) human shape are the same 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 9 Similarity 10 Isometry-invariant distance A monkey shape is Non-negative function more similar to a satisfying for all Similarity: deformation of a monkey shape are (In particular, than to a are -isometric; -isometric if only if ) Symmetry: human shape Triangle inequality: Corollary: is a metric on the quotient space Given discretized shapes sampled with radius Consistency to sampling: 11 Canonical forms distance Gromov-Hausdorff distance Minimum-distortion Minimum-distortion Isometric Isometric embedding embedding embedding embedding Gromov-Hausdorff distance: include Compute Hausdorff distance over all isometries in into minimization No fixed embedding space will give distortion-less canonical forms Mikhail Gromov 2 12

3 13 14 Properties of Gromov-Hausdorff distance Alternative definition I (metric coupling) Metric on the quotient space of isometries of shapes Similarity: are -isometric; are -isometric where is the disjoint union of Consistent to sampling: given discretized shapes the (semi-) metric satisfies sampled with radius Generalization of Hausdorff distance: Hausdorff distance between subsets of a metric space Gromov-Hausdorff distance between metric spaces Gromov, 1981 Mémoli, Alternative definition I (metric coupling) Correspondence A subset is called a correspondence between if for every there exists at least one such that similarly for every there exists such that Optimization over translates into finding the values of A lot of constraints! Particular case: given Mémoli, 2008 Correspondence distortion 17 Alternative definition II (correspondence distortion) 18 The distortion of correspondence is defined as Proof sketch 1. Show that for any there exists with Since, by definition of, are subspaces of some such that Let By triangle inequality, for 3

4 19 Alternative definition II (correspondence distortion) 2. Show that First, for any Let For each It is sufficient to show that there is a (semi-)metric such that Construct the metric, on the disjoint union Since for ). 21 Alternative definition III, it is straightforward 22 Alternative definition III is distorted by when embedded into measures how much 23 is distorted by when embedded into Generalized MDS Alternative definition III measures how far is is a (semi-)metric on We only need to show metric properties hold on measures how much, as follows On (in particular, for, Second, we need to show that from being the inverse of A. Bronstein, M. Bronstein & R. Kimmel, Alternative definition II (correspondence distortion) 24

5 25 26 Difficulties Local representation is sampled at represented as a triangular mesh. Any point falls into one of the triangles. How to represent points on? Within the triangle, it can be represented as convex combination Global parametrization is not always available. of triangle vertices, Some local representation is required in general case. No more closed-form expression for. Metric needs to be approximated. Minimization algorithm. Barycentric coordinates. We will need to hle discrete indices in minimization algorithm Geodesic distances Geodesic distance approximation Distance terms can be precomputed, since are fixed. How to compute distance terms? No more closed-form expression. Cannot be precomputed, since are minimization variables. can fall anywhere on the mesh. Precompute for all. Approximate for any. Approximation from. First order accurate: Consistent with data: Symmetric: Smoothness: is a closed-form expression for its derivatives is available to minimization algorithm. Might be only at some points or along some lines. Efficiently computed: constant complexity independent of Geodesic distance approximation Geodesic distance approximation Compute for. We have already encountered this problem in fast marching. falls into triangle is represented as Wavefront arrives at triangle vertex at time. When does it arrive to? Adopt planar wavefront model. Particular case: Hence, we can precompute distances Distance map is linear in the triangle (hence, linear in ) Solve for coefficients obtain a linear interpolant How to compute from? 5

6 31 Geodesic distance approximation General case: falls into triangle 32 A four-step dance is represented as Apply previous steps in triangle Apply once again in triangle to obtain to obtain Minimization algorithm Quadratic stress How to minimize the generalized stress? Quadratic stress Particular case: L2 stress Fix all all except for some Stress as a function of. only becomes quadratic Quadratic stress Closed-form solution for minimizer of 35 is positive semi-definite. is convex in (but not necessarily in together). 36 Minimization algorithm Initialize For each Fix Problem: solution might be outside the triangle. Solution: find constrained minimizer Select compute gradient corresponding to maximum Compute minimizer If constraints are active translate Closed-form solution still exists. 6 to adjacent triangle. Iterate until convergence.

7 37 38 How to move to adjacent triangles? Point on edge on edge opposite to. If edge is not shared by any other triangle we are on the boundary no translation. inside on edge on vertex Otherwise, express the point as in triangle. contains same values as. Three cases All : inside triangle. : on edge opposite to. : on vertex. May be permuted due to different vertex ordering in. Complication: is not on the edge. Evaluate gradient in. If points inside triangle, update to Point on vertex MDS vs GMDS on vertex. For each triangle sharing vertex Express point as in. Evaluate gradient in. Reject triangles with pointing outside. Select triangle with maximum. Update to. MDS Stress Analytic expression for Nonconvex problem Variables: Euclidean coordinates of the points Generalized MDS Generalized stress must be interpolated Nonconvex problem Variables: points on in barycentric coordinates Multiresolution Multiresolution Stress is non convex many small local minima. Initialize at the coarsest resolution in. Straightforward minimization gives poor results. How to initialize GMDS? For Starting at initialization, solve the GMDS problem Multiresolution: Create a hierarchy of grids in, Interpolate solution to next resolution level Each grid comprises Sampling: Geodesic distance matrix: Return. 7

8 43 44 Multiresolution encore GMDS Interpolation GMDS So far, we created a hierarchy of embedded spaces. One step further: create a hierarchy of embedding spaces Labeling problem Build a graph with vertices edges Label each vertex Minimum distortion correspondence = graph labeling problem MATLAB intermezzo Efficient solvers with good global convergence properties GMDS Complexity: Hierarchical solution complexity can be lowered to Torresani, Kolmogorov, Rother 2008 Wang, B 2010 Discrete Gromov-Hausdorff distance 47 Numerical example Two coupled GMDS problems Can be cast as a constrained problem Canonical forms (MDS based on 500 points) Bronstein, Bronstein & Kimmel, Bronstein, Bronstein & Kimmel, 2006 Gromov-Hausdorff distance (GMDS based on 50 points) 48

9 49 Extrinsic similarity using Gromov-Hausdorff distance EXTRINSIC SIMILARITY Connection to canonical form distance 50 Congruence Euclidean isometry ICP distance: GH distance with Euclidean metric: Connection between Euclidean GH ICP distances: Mémoli, 2008 Mémoli (2008) Correspondence again L p Gromov-Hausdorff distance A different representation for correspondence using indicator functions We can give an alternative formulation of the Gromov-Hausdorff distance defines a valid correspondence if Can we define an L p version of the Gromov-Hausdorff distance by relaxing the above definition? Measure coupling Gromov-Wasserstein distance Let be probability measures defined on The relaxed version of the Gromov-Hausdorff distance is given by (a metric space with measure is called a metric measure or mm space) A measure on is a coupling of if is referred to as Gromov-Wasserstein distance [Memoli 2007] for all measurable sets The measure can be considered as a relaxed version of the indicator function or as fuzzy correspondence Mémoli, 2007 Mémoli,

10 55 56 Earth Mover s distance The analogy Let be a metric space, measures supported on Define the coupling of The Wasserstein or Earth Mover s distance (EMD) is given by Hausdorff Distance between subsets of a metric space. Wasserstein Distance between subsets of a metric measure space. EMD as optimal mass transport: mass transported from to distance traveled Gromov-Hausdorff Distance between metric spaces Gromov-Wasserstein Distance between metric measure spaces Mémoli, 2007 Mémoli,