“Steklov Spectral Geometry for Extrinsic Shape Analysis” by Wang, Ben-Chen, Polterovich and Solomon

  • ©Yu Wang, Mirela (Miri) Ben-Chen, Iosif Polterovich, and Justin M. Solomon

Conference:


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Title:

    Steklov Spectral Geometry for Extrinsic Shape Analysis

Session/Category Title:   Maps and Operators


Presenter(s)/Author(s):



Abstract:


    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.

References:


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