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Steklov Spectral Geometry for Extrinsic Shape Analysis

Banff International Research Station for Mathematical Innovation and Discovery

Computer graphics and geometry processing study the representation, processing, and analysis of 3D shapes, with wide applications to brain imaging, computer vision, computer aided design and engineering, and so on. Intrinsic approaches, usually based on the Laplace-Beltrami operator, have been popular in computer graphics. However, intrinsic approaches cannot capture the spatial embedding of a shape up to rigid motion, and many previous extrinsic methods lack theoretical justification. Instead, we advocate using the Dirichlet-to-Neumann operator as an extrinsic alternative to the Laplacian for geometry processing and shape analysis. 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.

Mathematics; Numerical analysis; Computer science; Global analysis

video/mp4

Author affiliation: MIT

2019-03-20

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68991

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/