Steklov Spectral Geometry for Extrinsic Shape Analysis
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.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Dec 14, 2018
- Source ID
- 10.1145/3152156
Entities
People
- Iosif Polterovich
- Justin Solomon
- Mirela Ben-chen
- Yu Wang
Organizations
- Army Research Office
- Canada Excellence Research Chairs
- European Research Council
- Fonds de Recherche du Québec Nature et technologies
- Israel Science Foundation
- Massachusetts Institute of Technology
- National Science Foundation
- Natural Sciences and Engineering Research Council
- Technion – Israel Institute of Technology
- Thomas and Stacey Siebel Foundation
- Université de Montréal