期刊
GRAPHICAL MODELS
卷 74, 期 -, 页码 121-129出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.gmod.2012.03.009
关键词
Discrete heat kernel; Discrete Riemannian metric; Laplace-Beltrami operator; Legendre duality principle; Discrete curvature flow
资金
- ONR [N000140910228]
- Direct For Computer & Info Scie & Enginr
- Div Of Information & Intelligent Systems [0916235] Funding Source: National Science Foundation
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1221339] Funding Source: National Science Foundation
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. Published by Elsevier Inc.
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