Steklov Spectral Geometry for Extrinsic Shape Analysis

TL;DR

This paper introduces the Steklov spectral geometry approach using the Dirichlet-to-Neumann operator for extrinsic shape analysis, capturing volumetric geometry and enabling a straightforward shift from intrinsic methods.

Contribution

It presents a novel extrinsic spectral geometry method based on the Steklov eigenvalue problem, with a practical discretization pipeline for large-scale meshes.

Findings

The Dirichlet-to-Neumann operator encodes volumetric shape information.

The proposed method integrates seamlessly into existing geometry processing frameworks.

Efficient numerical schemes enable large-scale eigenproblem solutions.

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.