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

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

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

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!

Gram matrices A matrix of inner products of the form is called a Gram matrix Properties: (positive semidefinite) Jørgen Pedersen Gram (1850-1916)

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 (1903-1990) [Schoenberg, 1935]: Points with the metric can be isometrically embedded into a Euclidean space if and only if

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)

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)

MATLAB® intermezzo Classic MDS Canonical forms

Classical scaling example B 1 1 1 A B 1 1 2 D A C C 1 D A 1 B C D 2

Topological invariance Deformation Deformation +Topology

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

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

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

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

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

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)

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

Laplacian eigenmaps example Classic MDS Laplacian eigenmap

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)

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:

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

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)

Pierre Simon de Laplace Laplace-Beltrami Pierre Simon de Laplace (1749-1827) Eugenio Beltrami (1835-1899)

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:

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

Chladni’s experimental setup allowing to visualize acoustic waves To see the sound Ernst Chladni ['kladnɪ] (1715-1782) Chladni’s experimental setup allowing to visualize acoustic waves E. Chladni, Entdeckungen über die Theorie des Klanges

Chladni plates Patterns seen by Chladni are solutions to stationary Helmholtz equation Solutions of this equation are eigenfunction of Laplace-Beltrami operator

The first eigenfunctions of the Laplace-Beltrami operator

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

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

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.

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

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

“ ” 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?

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?

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

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

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

“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

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

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

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

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