EE Technion, Spring then. can be isometrically embedded into can be realized as a Gram matrix of rank, Properties:

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1 5/25/200 2 A mathematical exercise Assume points with the metric are isometrically embeddable into Then, there exists a canonical form such that for all Spectral methods We can also write Alexander & Michael Bronstein, Advanced topics in vision Michael Bronstein, 200 Processing and Analysis of Geometric Shapes tosca.cs.technion.ac.il/book EE Technion, Spring A mathematical exercise A mathematical exercise Since the canonical form is defined up to isometry, we can arbitrarily set Element of a matrix Element matrix of an Conclusion: if points then are isometrically embeddable into Note: can be defined in different ways! 5 6 Gram matrices Back to our problem A matrix of inner products of the form If points with the metric is called a Gram matrix can be isometrically embedded into, then can be realized as a Gram matrix of rank, Properties: which is positive semidefinite (positive semidefinite) Jørgen Pedersen Gram (850-96) A positive semidefinite matrix can be written as of rank Isaac Schoenberg ( ) giving the canonical form [Schoenberg, 935]: Points with the metric can be isometrically embedded into a Euclidean space if and only if

2 5/25/ Classic MDS Properties of classic MDS Usually, a shape is not isometrically embeddable into a Eucludean space, Nested dimensions: the first dimensions of an -dimensional implying that (has negative eignevalues) canonical form are equal to an -dimensional canonical form We can approximate by a Gram matrix of rank The error introduced by taking instead of can be quantified as Classic MDS minimizes the strain Keep m largest eignevalues Global optimization problem no local convergence Canonical form computed as Method known as classic MDS (or classical scaling) Requires computing a few largest eigenvalues of a real symmetric matrix, which can be efficiently solved numerically (e.g. Arnoldi and Lanczos) 9 0 Classical scaling example B A B D C A 2 C MATLAB intermezzo Classic MDS Canonical forms A B C D A 2 B C 2 D D 2 Topological invariance 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 Deformation Deformation +Topology 2

3 5/25/ Local methods Think globally, act locally David Brower Laplacian matrix Matrix formulation Recall stress derivation in LS-MDS Local criterion how far apart the embedding takes neighboring points where is an matrix with elements Global criterion is called the Laplacian matrix where has zero eigenvalue 5 6 Local methods Minimum eigenvalue problems Compute canonical form by solving the optimization problem Lets look at a simplified case: one-dimensional embedding Trivial solution ( collapse to a single point ): points can Introduce a constraint avoiding trivial solution Express the problem using eigendecomposition Geometric intuition: find a unit vector shortened the most by the action of the matrix 7 8 Minimum eigenvalue problems Laplacian eigenmaps Solution of the problem Compute the canonical form by finding the eigenvectors of smallest non-trivial is given as the smallest non-trivial eigenvectors of The smallest eigenvalue is zero and the corresponding eigenvector is constant (collapsing to a point) 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 3

4 5/25/ Laplacian eigenmaps example 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 points to neighboring points In the continuous case, we have a smooth map that maps neighboring on surface Classic MDS Laplacian eigenmap 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) 2 22 Continuous case Continuous analog of Laplacian eigenmaps By the Cauchy-Schwarz inequality Canonical form computed as the minimization problem implying that is small if is small: i.e., points close to are mapped close to where: Continuous local criterion: is the space of square-integrable functions on Continuous global criterion: We can rewrite Stokes theorem Laplace-Beltrami operator Laplace-Beltrami 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 ( ) Eugenio Beltrami ( ) 4

5 5/25/ Properties of Laplace-Beltrami operator Continuous vs discrete problem Let be smooth functions on the surface. Then the Continuous: 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: Discrete: Laplace-Beltrami operator Laplacian To see the sound Chladni plates Ernst Chladni ['kladnɪ] (75-782) Patterns seen by Chladni are solutions to stationary Helmholtz equation Chladni s experimental setup allowing to visualize acoustic waves Solutions of this equation are eigenfunction of Laplace-Beltrami operator E. Chladni, Entdeckungen über die Theorie des Klanges Laplace-Beltrami operator Laplace-Beltrami eigenfunctions The first eigenfunctions of the Laplace-Beltrami operator An eigenfunction of the Laplace-Beltrami operator computed on different deformations of the shape, showing the invariance of the Laplace-Beltrami operator to isometries 5

6 5/25/ Laplace-Beltrami spectrum Shape DNA Eigendecomposition of Laplace-Beltrami operator of a compact shape gives a discrete set of eigenvalues and eigenfunctions [Reuter et al. 2006]: use the Laplace-Beltrami spectrum isometry-invariant shape descriptor ( shape DNA ) as an Since the Laplace-Beltrami operator is symmetric, eigenfunctions form an orthogonal basis for The eigenvalues and eigenfunctions are isometry invariant Images: Reuter et al. Laplace-Beltrami spectrum Shape DNA Uniqueness of representation ISOMETRIC SHAPES ARE ISOSPECTRAL ARE ISOSPECTRAL SHAPES ISOMETRIC? Shape similarity using Laplace-Beltrami spectrum Images: Reuter et al. Can one hear the shape of the drum? 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 Mark Kac (94-984) Can we hear the metric? More prosaically: can one reconstruct the shape (up to an isometry) from its Laplace-Beltrami spectrum? 6

7 5/25/ One cannot hear the shape of the drum! Discrete Laplace-Beltrami operator [Gordon et al. 99]: 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 Counter-example of isospectral but not isometric shapes Discrete Laplace-Beltrami eigenfunctions Discrete vs discretized Find the discrete eigenfunctions of the Laplace-Beltrami operator by solving the generalized eigenvalue problem Continuous surface Laplace-Beltrami operator Continuous eigenfunctions and eigenvalues where is an matrix whose columns are the eigenfunctions is a diagonal matrix of corresponding 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 Levy 2006 Reuter, Biasotti, Giorgi, Patane & Spagnuolo 2009 Discrete Laplacian Discretized Laplacian FEM 4 42 Discrete Laplace-Beltrami operator Properties of discrete Laplace-Beltrami operator The discrete analog of the properties of the continuous Laplace-Betrami operator is Discrete Laplacian Cotangent weight 2 Symmetry: Locality: if are not directly connected Euclidean case: if is Euclidean plane, (umbrella operator); or valence of vertex (Tutte) sum of areas of triangles sharing vertex Positive semidefinite: In order for the discretization to be consistent,. Tutte 963; Zhang Pinkall 993; Meyer 2003 Convergence: solution of discrete PDE with of continuous PDE with for converges to the solution 7

8 5/25/ No free lunch Finite elements method Laplacian matrix we used in Laplacian eigenmaps does not converge to the continuous Laplace-Beltrami operator Eigendecomposition problem in the weak form for any smooth 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 Given a finite basis can be expanded as Write a system of equation spanning a subspace of posed as a generalized eigenvalue problem Reuter, Biasotti, Giorgi, Patane & Spagnuolo