A novel algorithm for volume-preserving parameterizations of 3-manifolds

Mei Heng Yueh*, Tiexiang Li, Wen Wei Lin, Shing Tung Yau

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

Abstract

Manifold parameterizations have been applied to various fields of commercial industries. Several efficient algorithms for the computation of triangular surface mesh parameterizations have been pro- posed 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.

Original languageEnglish
Pages (from-to)1071-1098
Number of pages28
JournalSIAM Journal on Imaging Sciences
Volume12
Issue number2
DOIs
Publication statusPublished - 2019

Keywords

  • Genus-zero closed surface
  • Manifold parameterization
  • Simply connected 3-manifold
  • Volume-preserving parameterization
  • Volumetric stretch energy minimization

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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