2A mathematical exercise Assume points with the metric are isometrically embeddable intoThen, there exists a canonical form such thatfor allWe can also write
3A mathematical exercise Since the canonical form is defined up toisometry, we can arbitrarily set
4A mathematical exercise Element ofa matrixElement of anmatrixConclusion: if points are isometrically embeddable into thenNote: can be defined in different ways!
5Gram matrices A matrix of inner products of the form is called a Gram matrixProperties:(positive semidefinite)Jørgen Pedersen Gram( )
6Back to our problem… If points with the metric can be isometrically embedded into , thencan be realized as a Gram matrix of rank ,which is positive semidefiniteA positive semidefinite matrix of rankcan be written asgiving the canonical formIsaac Schoenberg( )[Schoenberg, 1935]: Points with the metric can be isometrically embedded into a Euclidean space if and only if
7Keep m largest eignevalues Classic MDSUsually, a shape is not isometrically embeddable into a Eucludean space, implying that (has negative eignevalues)We can approximate by a Gram matrix of rankKeep m largest eignevaluesCanonical form computed asMethod known as classic MDS (or classical scaling)
8Properties of classic MDS Nested dimensions: the first dimensions of an dimensionalcanonical form are equal to an -dimensional canonical formThe error introduced by taking instead of can be quantified asClassic MDS minimizes the strainGlobal optimization problem – no local convergenceRequires computing a few largest eigenvalues of a real symmetric matrix,which can be efficiently solved numerically (e.g. Arnoldi and Lanczos)
12Local methodsMake the embedding preserve local properties of the shapeMap neighboring points to neighboring pointsIf , then is small. We want the corresponding distance in the embedding space to be small
13Think globally, act locally Local methods“”Think globally, act locallyDavid BrowerLocal criterion how far apart the embedding takes neighboring pointsGlobal criterionwhere
14Recall stress derivation Laplacian matrixRecall stress derivationin LS-MDSMatrix formulationwhere is an matrix with elementsis called the Laplacian matrixhas zero eigenvalue
15Introduce a constraint avoiding trivial solution Local methodsCompute canonical form by solving the optimization problemIntroduce a constraint avoiding trivial solutionTrivial solution ( ): points can collapse to a single point
16Minimum eigenvalue problems Lets look at a simplified case: one-dimensional embeddingExpress the problem using eigendecompositionGeometric intuition: find a unit vector shortened the most by the action of the matrix
17Minimum eigenvalue problems Solution of the problemis given as the smallest non-trivial eigenvectors ofThe smallest eigenvalue is zero and the corresponding eigenvector is constant (collapsing to a point)
18Laplacian eigenmapsCompute the canonical form by finding the smallest non-trivial eigenvectors ofMethod called Laplacian eigenmap [Belkin&Niyogi]is sparse (computational advantage for eigendecomposition)We need the lower part of the spectrum ofNested dimensions like in classic MDS
19Laplacian eigenmaps example Classic MDSLaplacian eigenmap
20Inner product on tangent space (metric tensor) Continuous caseConsider 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 pointsIn the continuous case, we have a smooth map on surfaceLet be a point on and be a point obtained by an infinitesimal displacement from by a vector in the tangent planeBy Taylor expansion,Inner product on tangent space (metric tensor)
21Continuous case By the Cauchy-Schwarz inequality implying that is small if is small: i.e., points close to are mapped close toContinuous local criterion:Continuous global criterion:
22Continuous analog of Laplacian eigenmaps Canonical form computed as the minimization problemwhere:is the space of square-integrable functions onWe can rewriteStokes theorem
23Laplace-Beltrami operator The operator is called Laplace-Beltrami operatorNote: we define Laplace-Beltrami operator with minus, unlike many booksLaplace-Beltrami operator is a generalization of Laplacian to manifoldsIn the Euclidean plane,In coordinate notationIntrinsic property of the shape (invariant to isometries)
24Pierre Simon de Laplace Laplace-BeltramiPierre Simon de Laplace( )Eugenio Beltrami( )
25Properties of Laplace-Beltrami operator Let be smooth functions on the surface Then the Laplace-Beltrami operator has the following propertiesConstant eigenfunction: for anySymmetry:Locality: is independent of for any pointsEuclidean case: if is Euclidean plane andthenPositive semidefinite:
26Laplace-Beltrami operator Continuous vs discrete problemContinuous:Laplace-Beltrami operatorDiscrete:Laplacian
27Chladni’s experimental setup allowing to visualize acoustic waves To see the soundErnst Chladni ['kladnɪ]( )Chladni’s experimental setup allowing to visualize acoustic wavesE. Chladni, Entdeckungen über die Theorie des Klanges
28Chladni platesPatterns seen by Chladni are solutions to stationary Helmholtz equationSolutions of this equation are eigenfunction of Laplace-Beltrami operator
29The first eigenfunctions of the Laplace-Beltrami operator
30Laplace-Beltrami eigenfunctions An eigenfunction of the Laplace-Beltrami operator computed ondifferent deformations of the shape, showing the invariance of theLaplace-Beltrami operator to isometries
31Laplace-Beltrami spectrum Eigendecomposition of Laplace-Beltrami operator of a compact shape gives a discrete set of eigenvalues and eigenfunctionsSince the Laplace-Beltrami operator is symmetric, eigenfunctionsform an orthogonal basis forThe eigenvalues and eigenfunctions are isometry invariant
32Laplace-Beltrami spectrum Shape DNA[Reuter et al. 2006]: use the Laplace-Beltrami spectrum as an isometry-invariant shape descriptor (“shape DNA”)Laplace-Beltrami spectrumImages: Reuter et al.
33Shape similarity using Laplace-Beltrami spectrum Shape DNAShape similarity using Laplace-Beltrami spectrumImages: Reuter et al.
34ISOMETRIC SHAPES ARE ISOSPECTRAL ARE ISOSPECTRAL SHAPES ISOMETRIC? Uniqueness of representationISOMETRIC SHAPES ARE ISOSPECTRALARE ISOSPECTRAL SHAPES ISOMETRIC?
35“ ” 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?
36To hear the shapeIn Chladni’s experiments, the spectrum describes acoustic characteristics of the plates (“modes” of vibrations)What can be “heard” from the spectrum:Total Gaussian curvatureEuler characteristicAreaCan we “hear” the metric?
37Counter-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
38Discrete Laplace-Beltrami operator Let the surface be sampled at points and represented as a triangular mesh , and letDiscrete version of the Laplace-Beltrami operatorIn matrix notationwhere
39Discrete Laplace-Beltrami eigenfunctions Find the discrete eigenfunctions of the Laplace-Beltrami operator by solving the generalized eigenvalue problemwhere is an matrix whose columns are the eigenfunctionsis a diagonal matrix of corresponding eigenvaluesLevy 2006Reuter, Biasotti, Giorgi, Patane & Spagnuolo 2009
40“Discretized Laplacian” Discrete vs discretizedContinuous surfaceLaplace-Beltrami operatorContinuous eigenfunctions and eigenvaluesDiscretize the surfaceas a graphDiscretize Laplace-Beltramioperator, preserving someof the continuous propertiesDiscretizeeigenfunctionsand eigenvaluesGraph LaplacianEigendecompositionEigendecomposition“Discrete Laplacian”“Discretized Laplacian”FEM
41Discrete Laplace-Beltrami operator Discrete Laplacian1Cotangent weight2(umbrella operator); orvalence of vertex (Tutte)sum of areas of trianglessharing vertex1. Tutte 1963; Zhang 20042. Pinkall 1993; Meyer 2003
42Properties of discrete Laplace-Beltrami operator The discrete analog of the properties of the continuous Laplace-Betrami operator isSymmetry:Locality: if are not directly connectedEuclidean case: if is Euclidean plane,Positive semidefinite:In order for the discretization to be consistent,Convergence: solution of discrete PDE with converges to the solutionof continuous PDE with for
43No free lunchLaplacian matrix we used in Laplacian eigenmaps does not converge to the continuous Laplace-Beltrami operatorThere 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
44Finite elements method Eigendecomposition problem in the weak formfor any smoothGiven a finite basis spanning a subspace ofcan be expanded asWrite a system of equationposed as a generalized eigenvalue problemReuter, Biasotti, Giorgi, Patane & Spagnuolo 2009