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.J. Solomon acknowledges funding from an MIT–Skoltech Seed Fund grant ("Boundary Element Methods for Shape Analysis"), the Skoltech–MIT Next Generation Program, the MIT Research Support Committee, an Amazon Research Award, Army Research Office grant W911NF-12-R-0011 ("Smooth Modeling of Flows on Graphs"), and National Science Foundation grant IIS-1838071 ("BIGDATA:F:Statistical and Computational Optimal Transport for Geometric Data Analysis"). I. Polterovich acknowledges the support of NSERC, FRQNT, the Canada Research Chairs program, and the Weston Visiting Professorship program of the Weizmann Institute of Science where part of this research was performed. M. Ben-Chen acknowledges funding from the European Research Council (ERC starting grant no. 714776 "OPREP") and the Israel Science Foundation (grant no. 504/16). Y. Wang acknowledges the support of the MIT Grier Presidential Fellowship and Siebel Scholarship