Steklov Geometry Processing: An Extrinsic Approach to Spectral Shape Analysis

by   Yu Wang, et al.

We propose Steklov geometry processing, an extrinsic approach to 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, while many previous extrinsic methods lack theoretical justification. Instead, we propose a systematic approach by considering 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 encodes the 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.


page 6

page 10

page 11

page 13

page 14

page 18


Steklov Spectral Geometry for Extrinsic Shape Analysis

We propose using the Dirichlet-to-Neumann operator as an extrinsic alter...

Shape-from-intrinsic operator

Shape-from-X is an important class of problems in the fields of geometry...

Surface Networks

We study data-driven representations for three-dimensional triangle mesh...

A Simple Approach to Intrinsic Correspondence Learning on Unstructured 3D Meshes

The question of representation of 3D geometry is of vital importance whe...

Hamiltonian operator for spectral shape analysis

Many shape analysis methods treat the geometry of an object as a metric ...

A Registration-free approach for Statistical Process Control of 3D scanned objects via FEM

Recent work in on-line Statistical Process Control (SPC) of manufactured...

WESD - Weighted Spectral Distance for Measuring Shape Dissimilarity

This article presents a new distance for measuring shape dissimilarity b...