SIAM Journal on Imaging Sciences, Volume 13, Issue 3, Page 1536-1564, January 2020.
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.

中文翻译：

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一种新的高效的属一3歧管参数化算法

SIAM影像科学杂志，第13卷，第3期，第1536-1564页，2020年1月。
流形的参数化已广泛应用于数值偏微分方程和计算机图形学领域。为此，近年来，由不同的研究小组开发了几种有效且可靠的数值算法，用于计算三角形和四面体网格参数化。但是，当歧管的拓扑不平凡时（例如，拓扑实体圆环的3个歧管），仍然具有挑战性。在本文中，我们提出了一种新颖的体积拉伸能量最小化算法，用于将单个边界映射到标准圆环的环形多面体的体积保持参数化。此外，该算法还可用于计算属一封闭曲面和标准圆环之间的等角映射。数值实验表明，该算法是有效的，并且在映射的双射性上表现良好。演示了在流形配准和分区上的应用，以显示我们算法的鲁棒性。