Steklov Spectral Geometry for Extrinsic Shape Analysis

We propose using the Dirichlet-to-Neumann operator as an extrinsic alternative to the Laplacian for spectral geometry processing and shape analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator, cannot capture the spatial embedding of a shape up to rigid motion, and many previous extrinsic methods lack theoretical justification. Instead, we consider the Steklov eigenvalue problem, computing the spectrum of the Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable property of this operator is that it completely encodes volumetric geometry. We use the boundary element method (BEM) to discretize the operator, accelerated by hierarchical numerical schemes and preconditioning; this pipeline allows us to solve eigenvalue and linear problems on large-scale meshes despite the density of the Dirichlet-to-Neumann discretization. We further demonstrate that our operators naturally fit into existing frameworks for geometry processing, making a shift from intrinsic to extrinsic geometry as simple as substituting the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.Comment: Additional experiments adde

Solving eigenvalue problems on curved surfaces using the Closest Point Method

Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace--Beltrami operator on rather general curved surfaces. Our algorithm, which is based on the Closest Point Method, relies on an embedding of the surface in a higher-dimensional space, where standard Cartesian finite difference and interpolation schemes can be easily applied. We show that there is a one-to-one correspondence between a problem defined in the embedding space and the original surface problem. For open surfaces, we present a simple way to impose Dirichlet and Neumann boundary conditions while maintaining second-order accuracy. Convergence studies and a series of examples demonstrate the effectiveness and generality of our approach

Image Segmentation with Eigenfunctions of an Anisotropic Diffusion Operator

We propose the eigenvalue problem of an anisotropic diffusion operator for image segmentation. The diffusion matrix is defined based on the input image. The eigenfunctions and the projection of the input image in some eigenspace capture key features of the input image. An important property of the model is that for many input images, the first few eigenfunctions are close to being piecewise constant, which makes them useful as the basis for a variety of applications such as image segmentation and edge detection. The eigenvalue problem is shown to be related to the algebraic eigenvalue problems resulting from several commonly used discrete spectral clustering models. The relation provides a better understanding and helps developing more efficient numerical implementation and rigorous numerical analysis for discrete spectral segmentation methods. The new continuous model is also different from energy-minimization methods such as geodesic active contour in that no initial guess is required for in the current model. The multi-scale feature is a natural consequence of the anisotropic diffusion operator so there is no need to solve the eigenvalue problem at multiple levels. A numerical implementation based on a finite element method with an anisotropic mesh adaptation strategy is presented. It is shown that the numerical scheme gives much more accurate results on eigenfunctions than uniform meshes. Several interesting features of the model are examined in numerical examples and possible applications are discussed

High frequency oscillations of first eigenmodes in axisymmetric shells as the thickness tends to zero

The lowest eigenmode of thin axisymmetric shells is investigated for two physical models (acoustics and elasticity) as the shell thickness (2$\epsilon$) tends to zero. Using a novel asymptotic expansion we determine the behavior of the eigenvalue $\lambda$($\epsilon$) and the eigenvector angular frequency k($\epsilon$) for shells with Dirichlet boundary conditions along the lateral boundary, and natural boundary conditions on the other parts. First, the scalar Laplace operator for acoustics is addressed, for which k($\epsilon$) is always zero. In contrast to it, for the Lam{\'e} system of linear elasticity several different types of shells are defined, characterized by their geometry, for which k($\epsilon$) tends to infinity as $\epsilon$ tends to zero. For two families of shells: cylinders and elliptical barrels we explicitly provide $\lambda$($\epsilon$) and k($\epsilon$) and demonstrate by numerical examples the different behavior as $\epsilon$ tends to zero

Analysis and discretization of the volume penalized Laplace operator with Neumann boundary conditions

We study the properties of an approximation of the Laplace operator with Neumann boundary conditions using volume penalization. For the one-dimensional Poisson equation we compute explicitly the exact solution of the penalized equation and quantify the penalization error. Numerical simulations using finite differences allow then to assess the discretisation and penalization errors. The eigenvalue problem of the penalized Laplace operator with Neumann boundary conditions is also studied. As examples in two space dimensions, we consider a Poisson equation with Neumann boundary conditions in rectangular and circular domains