Discrete Heat Kernel Determines Discrete Riemannian Metric Public Deposited

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This is the author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Elsevier and can be found at:  http://www.journals.elsevier.com/graphical-models/.

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  • 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.
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  • Zeng, W., Guo, R., Luo, F., & Gu, X. (2012). Discrete heat kernel determines discrete riemannian metric. Graphical Models, 74(4), 121-129. doi: 10.1016/j.gmod.2012.03.009
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  • description.provenance : Made available in DSpace on 2013-02-22T19:35:42Z (GMT). No. of bitstreams: 1 GuoRenMathematicsDiscreteHeatKernel.pdf: 1710279 bytes, checksum: 404cc57c05774734507957d453e52737 (MD5) Previous issue date: 2012-07
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