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Modal Shape Analysis beyond Laplacian(CAGP 2012)Klaus Hildebrandt, Christian Schulz, Christoph von Tycowicz, Konrad Polthier(brief) Presenter: ShiHao.WuHello, now I will briefly talk about a new spectral paper Modal Shape Analysis beyond Laplacian1Outline1, Quick review of Spectral2, Main contributions of the paper3, Results First I will do a quick review and then is the main contributions and results of the paper.2Chladni platesLet us see the vibration of the metal plate from the white sands.

I guess many one knew this famous physical phenomenone, Chladni plates, which allow us to see the vibration of the metal plate from the white sands3Chladni plates VS spetral

Left: In the late 1700's, the physicist Ernst Chladni was amazedby the patterns formed by sand on vibrating metal plates. Right: numericalsimulations obtained with a discretized Laplacian.The interesting is, somehow the Laplacian spectral in the right pictures can get some similar pattan like the Chladni 4How to transform to spetral space?In order to build eigenstructure of the surface.Discrete view:Define matrices on the graphs of surface, then find eigenvectors and eigenvalues. Continuous view:Define operators on the manifolds, then find eigenfunctions(eigenmodes) and eigenvalues.Generally speaking, in order to build the eigenstructure of the surface,we need to define some operators first.and the find the eigenfunctions. 5Laplace operator(Laplacian)

It turns out that,the Laplace operator is the most popular one.Beacuse Laplacian have good propoties, like the eigenfunctions of the Laplacian form an orthogonal basis. And the eigenfunctions are invariant under isometric deformations. However,this advantage leads to the disadvantage that Laplacian is insensitivity to extrinsic features of the surface, like sharp bends. 6Laplace operator(Laplacian)Advantage:eigenfunctions of the Laplacian form an orthogonal basis of the space of the surface(like Fourier basis)eigenvalues and eigenfunctions are invariant under isometric deformations of the surface

Disadvantage:insensitivity to extrinsic features of the surface, like sharp bendsIt turns out that,the Laplace operator is the most popular one.Beacuse Laplacian have good propoties, like the eigenfunctions of the Laplacian form an orthogonal basis. And the eigenfunctions are invariant under isometric deformations. However,this advantage leads to the disadvantage that Laplacian is insensitivity to extrinsic features of the surface, like sharp bends. 7Motivation and ContributionsNew operators derive from surface energy as alternatives to Lapacian, which is sensitive to features.Application to global mesh shape analysisvibration signature(descriptor): measures the similarity of points on a surfacefeature signature: identify features of surfaces

As a result, here comes to the motivation and contributions of this paper, that is, find new operators from surface energy as alternatives to Lapacian, which is sensitive to features.And the application is some signatures for geometry modal analysis8Whats surface energy?A twice continuously differentiable function of the surface.

(skip)At a minimum of E, the eigenmodes associated to the low eigenvalues of the Hessian of E, point into the direction that locally cause the least increase of energy.

Surface EnergyHessian on minimaEigenvaluesEigenmodesTo understand the paper,we should know that the surface energy is a twice continuously differentiable function define on the surface,so that we can compute the EigenvaluesOf the Hessian on the minima of the energy.9Example1 of surface energy(foundation of this paper)Dirichlet energy:

measure of how variable of , is intimately connected to Laplace basically.its Hessian equals the Laplace operator if the surface is a part of the Euclidean plane

A key example of surface energy is the Dirichlet energy, which measure how variable of the surface function sigma, and we can prove that,the Hessian of this energy is equals the Laplacian in somecircumstance. 10Example2 of surface energy(skip)(foundation of this paper)Deformation energy:

x is a surface mesh (the positions of the vertices) and measure angles, length of edges, or area of triangles. Example, thin shell simulation:

Another important energy like Dirichlet energy defined in the paper is the deformation energy, but we could skip that11Key observationLaplacian eigenvalue problem appears as theeigenvalue problem of the Hessian of the Dirichlet energy.So, By modifying the Dirichlet energy, we could get new operators better than Laplacian, using normal information

Next is the Key observation of this paper that Laplacian eigenvalue problem appears as theeigenvalue problem of the Hessian of the Dirichlet energy. So, By modifying the Dirichlet energy, we can get new operators better than Laplacian, using normal information of the surface

12Modified Dirichlet energy

Not only the idea, the modified equation is also simple, they just add this term, and kapa1 and kapa2 means the minima and maxima of the principal curvatures of the surface.13Discretization of Dirichlet energy(skip)

Next is the discretization,note that they only add the normal prodution.14Comparison and possible Explanation

Laplacian

But the results is quite different,here the left is the result of Laplacian eigenmode,and the right is the new eigenmode that is sensitive to the edges of the modal.One thing should be notice that different eigenvalues relate to different eigenmodes.15Comparison and possible ExplanationThe energy has its minimum at the origin of the space of normal vector fields.Therefore, eigenmodes of Hessian of that correspond to small eigenvalues have small function values in areas of high curvature, because then a variation in this direction causes less increase of energy

The author also give a possible explanation of these results that the .16Modes of Deformation energy(skip)Deformation energy:

So,I have talked the main part of this paper, and another part is about the deformation energy which I decide to skip17Eigen vibration modes from shell deformation energy(skip)

But we can see a short video of the Eigen vibration modes from shell deformation energy18Eigen vibration modes from shell deformation energy(skip)

But we can see a short video of the Eigen vibration modes from shell deformation energy19Vibration signature(skip)Based on the vibration modes of the surface with respect to the Discrete Shells energy

denote the eigenvalues and vibration eigenmodes of a mesh x. The eigenmodes with smaller eigenvalue receive higher weights and t is a parameter to control that

At last, they use the eigenvalues and eigenfunctions to define two signature,one is the Vibration signature20Vibration signature distancemeasures the similarity of points on a surface

Different distance thresholdWhich can measures the similarity of points on a surface21Feature signatureIdentify features of surfaces by normal information

Another signature is trying to locate the feature using the normal information22Vibration signatureVS Heat kernel signature

HKSV SThis picture show that the Heat kernel signature compare to Vibration signature. Which shows that the vibration signature is better Because in Heat kernel distance, the points on the knee is close to the points on the head. 23

Heat kernel signature VS Vibration signature

HKS

V Sdifferent tThis picture show that the Heat kernel signature compare to Vibration signature

24Feature signatureVS Heat kernel signature

HKSFSThis is the feature signature compare to the Heat kernal signature25Feature signatureVS Heat kernel signature

HKSFSEfficiency

Future workSpectral quadrangulations sensitive to feature

Create a subspace of the shape space of a mesh by vibration modes.(SIG 2012)

2829Application: use of eigenvectorsWhenever clustering is applicable, e.g., Mesh segmentation in spectral domain [Liu & Zhang 04, 05, 07]Surface reconstruction: grouping inside and outside tetrahedra [Kullori et al. 05]Shape correspondence: finding clusters of consistent or agreeable pair-wise matching [Leordeanu & Hebert 06]Favorite spectral projectMusic Visualization and editing

30Thank you !