A Novel Algorithm for Volume-Preserving Parameterizations of 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 Numerical Linear Algebra mathscidoc:1802.09001

SIAM Journal on Imaging Sciences, 12, (2), 1071–1098, 2019.6
Manifold parameterizations have been applied to various fields of commercial industries. Several efficient algorithms for the computation of triangular surface mesh parameterizations have been proposed in recent years. However, the computation of tetrahedral volumetric mesh parameterizations is more challenging due to the fact that the number of mesh points would become enormously large when the higher-resolution mesh is considered and the bijectivity of parameterizations is more difficult to guarantee. In this paper, we develop a novel volumetric stretch energy minimization algorithm for volume-preserving parameterizations of simply connected 3-manifolds with a single boundary under the restriction that the boundary is a spherical area-preserving mapping. In addition, our algorithm can also be applied to compute spherical angle- and area-preserving parameterizations of genus-zero closed surfaces, respectively. Several numerical experiments indicate that the developed algorithms are more efficient and reliable compared to other existing algorithms. Numerical results on applications of the manifold partition and the mesh processing for three-dimensional printing are demonstrated thereafter to show the robustness of the proposed algorithm.
quadratic energy minimization, genus zero closed surface, simply connected three-manifold, surface parameterization, volume-preserving parameterization
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  • https://doi.org/10.1137/18M1201184
  title={A Novel Algorithm for Volume-Preserving Parameterizations of 3-Manifolds},
  author={Mei-Heng Yueh, Tiexiang Li, Wen-Wei Lin, and Shing-Tung Yau},
  booktitle={SIAM Journal on Imaging Sciences},
Mei-Heng Yueh, Tiexiang Li, Wen-Wei Lin, and Shing-Tung Yau. A Novel Algorithm for Volume-Preserving Parameterizations of 3-Manifolds. 2019. Vol. 12. In SIAM Journal on Imaging Sciences. pp.1071–1098. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180228002240116376935.
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