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**Spectral methods 1 © Alexander & Michael Bronstein, 2006-2009**

tosca.cs.technion.ac.il/book Advanced topics in vision Processing and Analysis of Geometric Shapes EE Technion, Spring 2010 1

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**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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**A mathematical exercise**

Since the canonical form is defined up to isometry, we can arbitrarily set

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**A mathematical exercise**

Element of a matrix Element of an matrix Conclusion: if points are isometrically embeddable into then Note: can be defined in different ways!

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**Gram matrices A matrix of inner products of the form**

is called a Gram matrix Properties: (positive semidefinite) Jørgen Pedersen Gram ( )

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**Back to our problem… 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 Isaac Schoenberg ( ) [Schoenberg, 1935]: Points with the metric can be isometrically embedded into a Euclidean space if and only if

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**Keep m largest eignevalues**

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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**Properties of classic MDS**

Nested dimensions: the first dimensions of an dimensional canonical form are equal to an -dimensional canonical form The error introduced by taking instead of can be quantified as Classic MDS minimizes the strain 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)

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MATLAB® intermezzo Classic MDS Canonical forms

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**Classical scaling example**

B 1 1 1 A B 1 1 2 D A C C 1 D A 1 B C D 2

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**Topological invariance**

Deformation Deformation +Topology

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Local methods Make the embedding preserve local properties of the shape Map neighboring points to neighboring points If , then is small. We want the corresponding distance in the embedding space to be small

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**Think globally, act locally**

Local methods “ ” Think globally, act locally David Brower Local criterion how far apart the embedding takes neighboring points Global criterion where

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**Recall stress derivation**

Laplacian matrix Recall stress derivation in LS-MDS Matrix formulation where is an matrix with elements is called the Laplacian matrix has zero eigenvalue

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**Introduce a constraint avoiding trivial solution**

Local methods Compute canonical form by solving the optimization problem Introduce a constraint avoiding trivial solution Trivial solution ( ): points can collapse to a single point

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**Minimum eigenvalue problems**

Lets look at a simplified case: one-dimensional embedding Express the problem using eigendecomposition Geometric intuition: find a unit vector shortened the most by the action of the matrix

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**Minimum eigenvalue problems**

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)

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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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**Laplacian eigenmaps example**

Classic MDS Laplacian eigenmap

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**Inner product on tangent space (metric tensor)**

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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**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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**Continuous analog of Laplacian eigenmaps**

Canonical form computed as the minimization problem where: is the space of square-integrable functions on We can rewrite Stokes theorem

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**Laplace-Beltrami operator**

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

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**Pierre Simon de Laplace**

Laplace-Beltrami Pierre Simon de Laplace ( ) Eugenio Beltrami ( )

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**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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**Laplace-Beltrami operator**

Continuous vs discrete problem Continuous: Laplace-Beltrami operator Discrete: Laplacian

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**Chladni’s experimental setup allowing to visualize acoustic waves**

To see the sound Ernst Chladni ['kladnɪ] ( ) Chladni’s experimental setup allowing to visualize acoustic waves E. Chladni, Entdeckungen über die Theorie des Klanges

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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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**The first eigenfunctions of the Laplace-Beltrami operator**

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**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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**Laplace-Beltrami spectrum**

Eigendecomposition of Laplace-Beltrami operator of a compact shape gives a discrete set of eigenvalues and eigenfunctions Since the Laplace-Beltrami operator is symmetric, eigenfunctions form an orthogonal basis for The eigenvalues and eigenfunctions are isometry invariant

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**Laplace-Beltrami spectrum**

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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**Shape similarity using Laplace-Beltrami spectrum**

Shape DNA Shape similarity using Laplace-Beltrami spectrum Images: Reuter et al.

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**ISOMETRIC SHAPES ARE ISOSPECTRAL ARE ISOSPECTRAL SHAPES ISOMETRIC?**

Uniqueness of representation ISOMETRIC SHAPES ARE ISOSPECTRAL ARE ISOSPECTRAL SHAPES ISOMETRIC?

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**“ ” Can one hear the shape of the drum? Mark Kac (1914-1984)**

More prosaically: can one reconstruct the shape (up to an isometry) from its Laplace-Beltrami spectrum?

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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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**Counter-example of isospectral but not isometric shapes**

One cannot hear the shape of the drum! [Gordon et al. 1991]: Counter-example of isospectral but not isometric shapes

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**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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**Discrete Laplace-Beltrami eigenfunctions**

Find the discrete eigenfunctions of the Laplace-Beltrami operator by solving the generalized eigenvalue problem where is an matrix whose columns are the eigenfunctions is a diagonal matrix of corresponding eigenvalues Levy 2006 Reuter, Biasotti, Giorgi, Patane & Spagnuolo 2009

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**“Discretized Laplacian”**

Discrete vs discretized Continuous surface Laplace-Beltrami operator Continuous eigenfunctions and eigenvalues Discretize the surface as a graph Discretize Laplace-Beltrami operator, preserving some of the continuous properties Discretize eigenfunctions and eigenvalues Graph Laplacian Eigendecomposition Eigendecomposition “Discrete Laplacian” “Discretized Laplacian” FEM

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**Discrete Laplace-Beltrami operator**

Discrete Laplacian1 Cotangent weight2 (umbrella operator); or valence of vertex (Tutte) sum of areas of triangles sharing vertex 1. Tutte 1963; Zhang 2004 2. Pinkall 1993; Meyer 2003

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**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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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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**Finite elements method**

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 Reuter, Biasotti, Giorgi, Patane & Spagnuolo 2009

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