@article{b895f7d1b7a345079563d14d595155f0,
title = "A new efficient algorithm for volume-preserving parameterizations of genus-one 3-manifolds",
abstract = "Parameterizations of manifolds are widely applied to the fields of numerical partial differential equations and computer graphics. To this end, in recent years several efficient and reliable numerical algorithms have been developed by different research groups for the computation of triangular and tetrahedral mesh parameterizations. However, it is still challenging when the topology of manifolds is nontrivial, e.g., the 3-manifold of a topological solid torus. In this paper, we propose a novel volumetric stretch energy minimization algorithm for volume-preserving parameterizations of toroidal polyhedra with a single boundary being mapped to a standard torus. In addition, the algorithm can also be used to compute the equiareal mapping between a genus-one closed surface and the standard torus. Numerical experiments indicate that the developed algorithm is effective and performs well on the bijectivity of the mapping. Applications on manifold registrations and partitions are demonstrated to show the robustness of our algorithms.",
keywords = "Energy minimization, Toroidal polyhedral, Volume-preserving, Volumetric stretch energy",
author = "Yueh, {Mei Heng} and Tiexiang Li and Lin, {Wen Wei} and Yau, {Shing Tung}",
note = "Funding Information: \ast Received by the editors November 20, 2019; accepted for publication (in revised form) May 18, 2020; published electronically September 8, 2020. https://doi.org/10.1137/19M1301096 Funding: The work of the first author was partially supported by the Ministry of Science and Technology (MOST) grant 107-2115-M-003-012-MY2. The work of the second author was supported in part by the National Natural Science Foundation of China (NSFC) grant 11971105. The work of the third author was partially supported by the MOST grant 108-2119-M-009-006. The work of the authors was partially supported by the ST Yau Center in Taiwan, the Shing-Tung Yau Center at Southeast University, the Nanjing Center for Applied Mathematics (NCAM), and the Center of Mathematical Sciences and Applications (CMSA) at Harvard University. \dagger Department of Mathematics, National Taiwan Normal University, Taipei, 11677, Taiwan (yue@ntnu.edu.tw). \ddagger School of Mathematics, Southeast University, Nanjing 211189, People's Republic of China, and Nanjing Center for Applied Mathematics, Nanjing 211135, People's Republic of China (txli@seu.edu.cn). \S Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 300, Taiwan (wwlin@ math.nctu.edu.tw). \P Department of Mathematics, Harvard University, Cambridge, MA 02138 (yau@math.harvard.edu). Funding Information: The work of the first author was partially supported by the Ministry of Science and Technology (MOST) grant 107-2115-M-003-012-MY2. The work of the second author was supported in part by the National Natural Science Foundation of China (NSFC) grant 11971105. The work of the third author was partially supported by the MOST grant 108-2119-M-009-006. The work of the authors was partially supported by the ST Yau Center in Taiwan, the Shing-Tung Yau Center at Southeast University, the Nanjing Center for Applied Mathematics (NCAM), and the Center of Mathematical Sciences and Applications (CMSA) at Harvard University. Publisher Copyright: {\textcopyright} by SIAM.",
year = "2020",
doi = "10.1137/19M1301096",
language = "English",
volume = "13",
pages = "1536--1564",
journal = "SIAM Journal on Imaging Sciences",
issn = "1936-4954",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "3",
}