Discrete heat kernel determines discrete Riemannian metric

Wei Zeng, Ren Guo, Feng Luo, Xianfeng Gu

Research output: Contribution to journalArticlepeer-review

26 Scopus citations

Abstract

The Laplace-Beltrami operator of a smooth Riemannian manifold is determined by the Riemannian metric. Conversely, the heat kernel constructed from the eigenvalues and eigenfunctions of the Laplace-Beltrami operator determines the Riemannian metric. This work proves the analogy on Euclidean polyhedral surfaces (triangle meshes), that the discrete heat kernel and the discrete Riemannian metric (unique up to a scaling) are mutually determined by each other. Given a Euclidean polyhedral surface, its Riemannian metric is represented as edge lengths, satisfying triangle inequalities on all faces. The Laplace-Beltrami operator is formulated using the cotangent formula, where the edge weight is defined as the sum of the cotangent of angles against the edge. We prove that the edge lengths can be determined by the edge weights unique up to a scaling using the variational approach. The constructive proof leads to a computational algorithm that finds the unique metric on a triangle mesh from a discrete Laplace-Beltrami operator matrix.

Original languageEnglish (US)
Pages (from-to)121-129
Number of pages9
JournalGraphical Models
Volume74
Issue number4
DOIs
StatePublished - Jul 2012

All Science Journal Classification (ASJC) codes

  • Software
  • Modeling and Simulation
  • Geometry and Topology
  • Computer Graphics and Computer-Aided Design

Keywords

  • Discrete Riemannian metric
  • Discrete curvature flow
  • Discrete heat kernel
  • Laplace-Beltrami operator
  • Legendre duality principle

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