A New Efficient Algorithm for Volume-Preserving Parameterizations of Genus-One 3-Manifolds

Mei-Heng Yueh National Taiwan Normal University Tiexiang Li Southeast University Wen-Wei Lin National Chiao Tung University Shing-Tung Yau Harvard University

Computational Geometry mathscidoc:1911.09002

SIAM Journal on Imaging Sciences
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.
volumetric stretch energy, energy minimization, volume-preserving, toroidal polyhedra
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  • This paper has been accepted for publication in SIAM Journal on Imaging Sciences.
@inproceedings{mei-henga,
  title={A New Efficient Algorithm for Volume-Preserving Parameterizations of Genus-One 3-Manifolds},
  author={Mei-Heng Yueh, Tiexiang Li, Wen-Wei Lin, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191122181949651478513},
  booktitle={SIAM Journal on Imaging Sciences},
}
Mei-Heng Yueh, Tiexiang Li, Wen-Wei Lin, and Shing-Tung Yau. A New Efficient Algorithm for Volume-Preserving Parameterizations of Genus-One 3-Manifolds. In SIAM Journal on Imaging Sciences. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191122181949651478513.
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