Numerical geometry of nonrigid shapes Spectral embedding 1

Numerical geometry of non-rigid shapes Spectral embedding 1 Spectral embedding Lecture 6 © Alexander & Michael Bronstein tosca. cs. technion. ac. il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009

Numerical geometry of non-rigid shapes Spectral embedding 2 A mathematical exercise Assume points with the metric embeddable into Then, there exists a canonical form for all We can also write such that are isometrically

Numerical geometry of non-rigid shapes Spectral embedding A mathematical exercise Since the canonical form is defined up to isometry, we can arbitrarily set 3

Numerical geometry of non-rigid shapes Spectral embedding A mathematical exercise Element of a matrix Conclusion: if points Element of an matrix are isometrically embeddable into then Note: can be defined in different ways! 4

Numerical geometry of non-rigid shapes Spectral embedding 5 Gram matrices A matrix of inner products of the form is called a Gram matrix Properties: n n (positive semidefinite) Jørgen Pedersen Gram (1850 -1916)

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

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

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

Numerical geometry of non-rigid shapes Spectral embedding ® MATLAB intermezzo Classic MDS Canonical forms 9

10 Numerical geometry of non-rigid shapes Spectral embedding Classical scaling example B 1 1 1 A B 1 1 D C 1 A 2 C D A A B C D 1 2 1 B 1 1 1 C 2 1 D 1 1

Numerical geometry of non-rigid shapes Spectral embedding Local methods Make the embedding preserve local properties of the shape Map neighboring points to neighboring points If distance , then is small. We want the corresponding in the embedding space to be small 11

12 Numerical geometry of non-rigid shapes Spectral embedding Local methods “ Think globally, act locally ” David Brower Local criterion how far apart the embedding takes neighboring points Global criterion where

13 Numerical geometry of non-rigid shapes Spectral embedding Laplacian matrix Recall stress derivation in LS-MDS Matrix formulation where is an matrix with elements is called the Laplacian matrix n n has zero eigenvalue

Numerical geometry of non-rigid shapes Spectral embedding 14 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

Numerical geometry of non-rigid shapes Spectral embedding 15 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

Numerical geometry of non-rigid shapes Spectral embedding 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) 16

Numerical geometry of non-rigid shapes Spectral embedding Laplacian eigenmaps Compute the canonical form by finding the smallest non-trivial eigenvectors of Method called Laplacian eigenmap [Belkin&Niyogi] n is sparse (computational advantage for eigendecomposition) n We need the lower part of the spectrum of n Nested dimensions like in classic MDS 17

Numerical geometry of non-rigid shapes Spectral embedding Laplacian eigenmaps example Classic MDS Laplacian eigenmap 18

19 Numerical geometry of non-rigid shapes Spectral embedding 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 Let be a point on displacement from and on surface be a point obtained by an infinitesimal by a vector in the tangent plane By Taylor expansion, Inner product on tangent space (metric tensor)

Numerical geometry of non-rigid shapes Spectral embedding Continuous case By the Cauchy-Schwarz inequality implying that close to is small if are mapped close to Continuous local criterion: Continuous global criterion: is small: i. e. , points 20

Numerical geometry of non-rigid shapes Spectral embedding 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 21

Numerical geometry of non-rigid shapes Spectral embedding 22 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)

Numerical geometry of non-rigid shapes Spectral embedding Laplace-Beltrami Pierre Simon de Laplace (1749 -1827) Eugenio Beltrami (1835 -1899) 23

24 Numerical geometry of non-rigid shapes Spectral embedding Properties of Laplace-Beltrami operator Let be smooth functions on the surface . Then the Laplace-Beltrami operator has the following properties n Constant eigenfunction: for any n Symmetry: n Locality: n Euclidean case: if is independent of for any points is Euclidean plane and then n Positive semidefinite:

Numerical geometry of non-rigid shapes Spectral embedding Continuous vs discrete problem Continuous: Laplace-Beltrami operator Discrete: Laplacian 25

Numerical geometry of non-rigid shapes Spectral embedding 26 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

Numerical geometry of non-rigid shapes Spectral embedding 27 Chladni plates Patterns seen by Chladni are solutions to stationary Helmholtz equation Solutions of this equation are eigenfunction of Laplace-Beltrami operator

Numerical geometry of non-rigid shapes Spectral embedding Laplace-Beltrami operator The first eigenfunctions of the Laplace-Beltrami operator 28

Numerical geometry of non-rigid shapes Spectral embedding 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 29

Numerical geometry of non-rigid shapes Spectral embedding 30 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

31 Numerical geometry of non-rigid shapes Spectral embedding Shape DNA [Reuter et al. 2006]: use the Laplace-Beltrami spectrum isometry-invariant shape descriptor (“shape DNA”) Images: Reuter et al. Laplace-Beltrami spectrum as an

Numerical geometry of non-rigid shapes Spectral embedding Shape DNA Shape similarity using Laplace-Beltrami spectrum Images: Reuter et al. 32

Numerical geometry of non-rigid shapes Spectral embedding Uniqueness of representation ISOMETRIC SHAPES ARE ISOSPECTRAL SHAPES ISOMETRIC? 33

34 Numerical geometry of non-rigid shapes Spectral embedding “ 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? ”

Numerical geometry of non-rigid shapes Spectral embedding 35 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: n Total Gaussian curvature n Euler characteristic n Area Can we “hear” the metric?

Numerical geometry of non-rigid shapes Spectral embedding One cannot hear the shape of the drum! [Gordon et al. 1991]: Counter-example of isospectral but not isometric shapes 36

Numerical geometry of non-rigid shapes Spectral embedding 37 GPS embedding The eigenvalues and the eigenfunctions of the Laplace-Beltrami operator uniquely determine the metric tensor of the shape I. e. , one can recover the shape up to an isometry from [Rustamov, 2007]: Global Point Signature (GPS) embedding n An infinite-dimensional canonical form n Unique (unlike MDS-based canonical form, defined up to isometry) n Must be truncated for practical computation

Numerical geometry of non-rigid shapes Spectral embedding 38 Discrete Laplace-Beltrami operator Let the surface triangular mesh be sampled at points and represented as a , and let Discrete version of the Laplace-Beltrami operator Can be expressed as a matrix Discrete analog of constant eigenfunction property is satisfied by definition

Numerical geometry of non-rigid shapes Spectral embedding 39 Discrete vs discretized Continuous surface Laplace-Beltrami operator Discretize the surface Discretize Laplace-Beltrami operator, preserving some of the continuous properties Construct graph Laplacian Discrete Laplace-Beltrami operator Discretized Laplace-Beltrami operator

Numerical geometry of non-rigid shapes Spectral embedding 40 Properties of discrete Laplace-Beltrami operator The discrete analog of the properties of the continuous Laplace-Betrami operator is n Symmetry: n Locality: n Euclidean case: if if are not directly connected is Euclidean plane, n Positive semidefinite: In order for the discretization to be consistent, n Convergence: solution of discrete PDE with of continuous PDE with for converges to the solution

Numerical geometry of non-rigid shapes Spectral embedding 41 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, 2007]: there is no discretization of the Laplace-Beltrami operator satisfying simultaneously all the desired properties