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HOT: Hodge-optimized triangulations

Mullen, Patrick and Memari, Pooran and de Goes, Fernando and Desbrun, Mathieu (2011) HOT: Hodge-optimized triangulations. ACM Transactions on Graphics, 30 (4). Art. No. 103. ISSN 0730-0301.

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We introduce Hodge-optimized triangulations (HOT), a family of well-shaped primal-dual pairs of complexes designed for fast and accurate computations in computer graphics. Previous work most commonly employs barycentric or circumcentric duals; while barycentric duals guarantee that the dual of each simplex lies within the simplex, circumcentric duals are often preferred due to the induced orthogonality between primal and dual complexes. We instead promote the use of weighted duals ("power diagrams"). They allow greater flexibility in the location of dual vertices while keeping primal-dual orthogonality, thus providing a valuable extension to the usual choices of dual by only adding one additional scalar per primal vertex. Furthermore, we introduce a family of functionals on pairs of complexes that we derive from bounds on the errors induced by diagonal Hodge stars, commonly used in discrete computations. The minimizers of these functionals, called HOT meshes, are shown to be generalizations of Centroidal Voronoi Tesselations and Optimal Delaunay Triangulations, and to provide increased accuracy and flexibility for a variety of computational purposes.

Item Type:Article
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Desbrun, Mathieu0000-0003-3424-6079
Additional Information:© 2011 ACM. The authors thank David Cohen-Steiner and Omid Amini for early support and feedback, as well as Pierre Alliez for data. This research was partially funded through NSF grants (CCF-0811373, CMMI-0757106, and CCF-1011944), and by the generous support of Pixar Animation Studios.
Funding AgencyGrant Number
Pixar Animation StudiosUNSPECIFIED
Subject Keywords:Optimal triangulations; Discrete Exterior Calculus; Discrete Hodge Star; Optimal Transport
Issue or Number:4
Record Number:CaltechAUTHORS:20120104-094536122
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Official Citation:Mullen, P., Memari, P., de Goes, F., Desbrun, M. 2011. HOT: Hodge-Optimized Triangulations. ACM Trans. Graph. 30, 4, Article 103 (July 2011), 11 pages. DOI = 10.1145/1964921.1964998
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:28640
Deposited By: Tony Diaz
Deposited On:06 Jan 2012 17:16
Last Modified:03 Mar 2020 13:01

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