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1 Numerical Geometry of Non-Rigid Shapes Spectral methods Spectral methods © Alexander & Michael Bronstein, 2006-2009 © Michael Bronstein, 2010 tosca.cs.technion.ac.il/book 048921 Advanced topics in vision Processing and Analysis of Geometric Shapes EE Technion, Spring 2010

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2 Numerical Geometry of Non-Rigid Shapes Spectral methods A mathematical exercise Assume points with the metric are isometrically embeddable into Then, there exists a canonical form such that for all We can also write

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3 Numerical Geometry of Non-Rigid Shapes Spectral methods A mathematical exercise Since the canonical form is defined up to isometry, we can arbitrarily set

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4 Numerical Geometry of Non-Rigid Shapes Spectral methods A mathematical exercise Conclusion: if points are isometrically embeddable into then Element of a matrix Element of an matrix Note: can be defined in different ways!

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5 Numerical Geometry of Non-Rigid Shapes Spectral methods Gram matrices A matrix of inner products of the form is called a Gram matrix Jørgen Pedersen Gram (1850-1916) Properties: (positive semidefinite)

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6 Numerical Geometry of Non-Rigid Shapes Spectral methods Back to our problem… Isaac Schoenberg (1903-1990) [Schoenberg, 1935]: Points with the metric can be isometrically embedded into a Euclidean space if and only if If points with the metric can be isometrically embedded into, then can be realized as a Gram matrix of rank, which is positive semidefinite A positive semidefinite matrix of rank can be written as giving the canonical form

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7 Numerical Geometry of Non-Rigid Shapes Spectral methods Classic MDS Usually, a shape is not isometrically embeddable into a Eucludean space, implying that (has negative eignevalues) We can approximate by a Gram matrix of rank Keep m largest eignevalues Canonical form computed as Method known as classic MDS (or classical scaling)

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8 Numerical Geometry of Non-Rigid Shapes Spectral methods Properties of classic MDS Nested dimensions: the first dimensions of an -dimensional canonical form are equal to an -dimensional canonical form Global optimization problem – no local convergence Requires computing a few largest eigenvalues of a real symmetric matrix, which can be efficiently solved numerically (e.g. Arnoldi and Lanczos) The error introduced by taking instead of can be quantified as Classic MDS minimizes the strain

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9 Numerical Geometry of Non-Rigid Shapes Spectral methods MATLAB ® intermezzo Classic MDS Canonical forms

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10 Numerical Geometry of Non-Rigid Shapes Spectral methods Classical scaling example 1 B D A C 1 1 1 1 B A C 2 A A1 BCD B C D 21 111 211 111 D 1

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11 Numerical Geometry of Non-Rigid Shapes Spectral methods Deformation +Topology Topological invariance

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12 Numerical Geometry of Non-Rigid Shapes Spectral methods Local methods Make the embedding preserve local properties of the shape If, then is small. We want the corresponding distance in the embedding space to be small Map neighboring points to neighboring points

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13 Numerical Geometry of Non-Rigid Shapes Spectral methods Local methods Think globally, act locally Local criterion how far apart the embedding takes neighboring points “ ” David Brower Global criterion where

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14 Numerical Geometry of Non-Rigid Shapes Spectral methods Laplacian matrix where is an matrix with elements Matrix formulation Recall stress derivation in LS-MDS is called the Laplacian matrix has zero eigenvalue

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15 Numerical Geometry of Non-Rigid Shapes Spectral methods Local methods Compute canonical form by solving the optimization problem Trivial solution ( ): points can collapse to a single point Introduce a constraint avoiding trivial solution

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16 Numerical Geometry of Non-Rigid Shapes Spectral methods Minimum eigenvalue problems Lets look at a simplified case: one-dimensional embedding Geometric intuition: find a unit vector shortened the most by the action of the matrix Express the problem using eigendecomposition

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17 Numerical Geometry of Non-Rigid Shapes Spectral methods Solution of the problem is given as the smallest non-trivial eigenvectors of The smallest eigenvalue is zero and the corresponding eigenvector is constant (collapsing to a point) Minimum eigenvalue problems

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18 Numerical Geometry of Non-Rigid Shapes Spectral methods Laplacian eigenmaps Compute the canonical form by finding the smallest non-trivial eigenvectors of Method called Laplacian eigenmap [Belkin&Niyogi] is sparse (computational advantage for eigendecomposition) We need the lower part of the spectrum of Nested dimensions like in classic MDS

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19 Numerical Geometry of Non-Rigid Shapes Spectral methods Laplacian eigenmaps example Classic MDSLaplacian eigenmap

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20 Numerical Geometry of Non-Rigid Shapes Spectral methods Continuous case Consider a one-dimensional embedding (due to nested dimension property, each dimension can be considered separately) We were trying to find a map that maps neighboring points to neighboring points In the continuous case, we have a smooth map on surface Let be a point on and be a point obtained by an infinitesimal displacement from by a vector in the tangent plane By Taylor expansion, Inner product on tangent space (metric tensor)

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21 Numerical Geometry of Non-Rigid Shapes Spectral methods Continuous case By the Cauchy-Schwarz inequality implying that is small if is small: i.e., points close to are mapped close to Continuous local criterion: Continuous global criterion:

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22 Numerical Geometry of Non-Rigid Shapes Spectral methods Continuous analog of Laplacian eigenmaps Canonical form computed as the minimization problem where: Stokes theorem We can rewrite is the space of square-integrable functions on

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23 Numerical Geometry of Non-Rigid Shapes Spectral methods Laplace-Beltrami operator The operator is called Laplace-Beltrami operator Laplace-Beltrami operator is a generalization of Laplacian to manifolds In the Euclidean plane, Intrinsic property of the shape (invariant to isometries) Note: we define Laplace-Beltrami operator with minus, unlike many books In coordinate notation

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24 Numerical Geometry of Non-Rigid Shapes Spectral methods Laplace-Beltrami Pierre Simon de Laplace (1749-1827) Eugenio Beltrami (1835-1899)

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25 Numerical Geometry of Non-Rigid Shapes Spectral methods Properties of Laplace-Beltrami operator Let be smooth functions on the surface. Then the Laplace-Beltrami operator has the following properties Constant eigenfunction: for any Symmetry: Locality: is independent of for any points Euclidean case: if is Euclidean plane and then Positive semidefinite:

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26 Numerical Geometry of Non-Rigid Shapes Spectral methods Continuous vs discrete problem Continuous: Discrete: Laplace-Beltrami operator Laplacian

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27 Numerical Geometry of Non-Rigid Shapes Spectral methods To see the sound Chladni’s experimental setup allowing to visualize acoustic waves Ernst Chladni ['kladn ɪ ] (1715-1782) E. Chladni, Entdeckungen über die Theorie des Klanges

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28 Numerical Geometry of Non-Rigid Shapes Spectral methods Chladni plates Patterns seen by Chladni are solutions to stationary Helmholtz equation Solutions of this equation are eigenfunction of Laplace-Beltrami operator

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29 Numerical Geometry of Non-Rigid Shapes Spectral methods The first eigenfunctions of the Laplace-Beltrami operator Laplace-Beltrami operator

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30 Numerical Geometry of Non-Rigid Shapes Spectral methods Laplace-Beltrami eigenfunctions An eigenfunction of the Laplace-Beltrami operator computed on different deformations of the shape, showing the invariance of the Laplace-Beltrami operator to isometries

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31 Numerical Geometry of Non-Rigid Shapes Spectral methods Laplace-Beltrami spectrum Eigendecomposition of Laplace-Beltrami operator of a compact shape gives a discrete set of eigenvalues and eigenfunctions The eigenvalues and eigenfunctions are isometry invariant Since the Laplace-Beltrami operator is symmetric, eigenfunctions form an orthogonal basis for

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32 Numerical Geometry of Non-Rigid Shapes Spectral methods Shape DNA [Reuter et al. 2006]: use the Laplace-Beltrami spectrum as an isometry-invariant shape descriptor (“shape DNA”) Laplace-Beltrami spectrum Images: Reuter et al.

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33 Numerical Geometry of Non-Rigid Shapes Spectral methods Shape DNA Shape similarity using Laplace-Beltrami spectrum Images: Reuter et al.

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34 Numerical Geometry of Non-Rigid Shapes Spectral methods Uniqueness of representation ISOMETRIC SHAPES ARE ISOSPECTRAL ARE ISOSPECTRAL SHAPES ISOMETRIC?

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35 Numerical Geometry of Non-Rigid Shapes Spectral methods Mark Kac (1914-1984) Can one hear the shape of the drum? “ ” More prosaically: can one reconstruct the shape (up to an isometry) from its Laplace-Beltrami spectrum?

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36 Numerical Geometry of Non-Rigid Shapes Spectral methods To hear the shape In Chladni’s experiments, the spectrum describes acoustic characteristics of the plates (“modes” of vibrations) What can be “heard” from the spectrum: Total Gaussian curvature Euler characteristic Area Can we “hear” the metric?

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37 Numerical Geometry of Non-Rigid Shapes Spectral methods One cannot hear the shape of the drum! [Gordon et al. 1991]: Counter-example of isospectral but not isometric shapes

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38 Numerical Geometry of Non-Rigid Shapes Spectral methods Discrete Laplace-Beltrami operator Let the surface be sampled at points and represented as a triangular mesh, and let Discrete version of the Laplace-Beltrami operator In matrix notation where

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39 Numerical Geometry of Non-Rigid Shapes Spectral methods Find the discrete eigenfunctions of the Laplace-Beltrami operator by solving the generalized eigenvalue problem Levy 2006 Reuter, Biasotti, Giorgi, Patane & Spagnuolo 2009 where is an matrix whose columns are the eigenfunctions is a diagonal matrix of corresponding eigenvalues Discrete Laplace-Beltrami eigenfunctions

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40 Numerical Geometry of Non-Rigid Shapes Spectral methods Discrete vs discretized Continuous surface Laplace-Beltrami operator Discretize the surface as a graph Discretize Laplace-Beltrami operator, preserving some of the continuous properties Graph Laplacian Eigendecomposition Continuous eigenfunctions and eigenvalues Eigendecomposition Discretize eigenfunctions and eigenvalues “Discrete Laplacian”“Discretized Laplacian”FEM

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41 Numerical Geometry of Non-Rigid Shapes Spectral methods 1. Tutte 1963; Zhang 2004 2. Pinkall 1993; Meyer 2003 Cotangent weight 2 sum of areas of triangles sharing vertex Discrete Laplacian 1 (umbrella operator); or valence of vertex (Tutte) Discrete Laplace-Beltrami operator

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42 Numerical Geometry of Non-Rigid Shapes Spectral methods Properties of discrete Laplace-Beltrami operator The discrete analog of the properties of the continuous Laplace-Betrami operator is Symmetry: Locality: if are not directly connected Euclidean case: if is Euclidean plane, Positive semidefinite: In order for the discretization to be consistent, Convergence: solution of discrete PDE with converges to the solution of continuous PDE with for

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43 Numerical Geometry of Non-Rigid Shapes Spectral methods No free lunch Laplacian matrix we used in Laplacian eigenmaps does not converge to the continuous Laplace-Beltrami operator There exist many other approximations of the Laplace-Beltrami operator, satisfying different properties [Wardetzky et al. 2007]: there is no discretization of the Laplace- Beltrami operator satisfying simultaneously all the desired properties

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44 Numerical Geometry of Non-Rigid Shapes Spectral methods Finite elements method Reuter, Biasotti, Giorgi, Patane & Spagnuolo 2009 Eigendecomposition problem in the weak form for any smooth Given a finite basis spanning a subspace of can be expanded as Write a system of equation posed as a generalized eigenvalue problem

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