Modal Shape Analysis beyond Laplacian (CAGP 2012) Klaus Hildebrandt, Christian Schulz, Christoph von Tycowicz, Konrad Polthier (brief) Presenter: ShiHao.Wu

Outline 1, Quick review of Spectral 2, Main contributions of the paper 3, Results

Chladni plates Let us “see” the vibration of the metal plate from the white sands.

Chladni plates VS spetral Left: In the late 1700's, the physicist Ernst Chladni was amazed by the patterns formed by sand on vibrating metal plates. Right: numerical simulations obtained with a discretized Laplacian.

How 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.

Laplace 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 bends

Motivation and Contributions New operators derive from surface energy as alternatives to Lapacian, which is sensitive to features. Application to global mesh shape analysis – vibration signature(descriptor): measures the similarity of points on a surface – feature signature: identify features of surfaces

What’s 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 Energy Hessian on minima Eigenvalues Eigenmodes

Example1 of surface energy (foundation of this paper) Dirichlet energy: – measure of how variable of σ, is intimately connected to Laplace basically. – its Hessian equals the Laplace operator if the surface is a part of the Euclidean plane

Example2 of surface energy(skip) (foundation of this paper) Deformation energy: – x is a surface mesh (the positions of the vertices) – ƒ and ω measure angles, length of edges, or area of triangles. Example, thin shell simulation:

Key observation Laplacian eigenvalue problem appears as the eigenvalue 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

Modified Dirichlet energy

Discretization of Dirichlet energy(skip)

Comparison and possible Explanation Laplacian

Comparison and possible Explanation The 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

Modes of Deformation energy(skip) Deformation energy:

Eigen vibration modes from shell deformation energy(skip)

Vibration 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

Vibration signature distance measures the similarity of points on a surface Different distance threshold

Feature signature Identify features of surfaces by normal information

Vibration signature VS Heat kernel signature HKS V S

Heat kernel signature VS Vibration signature HKS V S different t

Feature signature VS Heat kernel signature HKS FS

Feature signature VS Heat kernel signature HKS FS

Efficiency

Future work Spectral quadrangulations sensitive to feature Create a subspace of the shape space of a mesh by vibration modes.(SIG 2012)

29 Application: use of eigenvectors Whenever 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 project Music Visualization and editing

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