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arXiv · 2402.00380

$n$-Dimensional Volumetric Stretch Energy Minimization for Volume-/Mass-Preserving Parameterizations

Zhong-Heng Tan·Tiexiang Li·Wen-Wei Lin·Shing-Tung Yau
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Abstract

In this paper, we develop an $n$ dimensional volumetric stretch energy ($n$-VSE) functional for the volume-/mass-preserving parameterization of the $n$-manifolds topologically equivalent to $n$-ball. The $n$-VSE has a lower bound and equal to it if and only if the map is volume-/mass-preserving. This motivates us to minimize the $n$-VSE to achieve the ideal volume-/mass-preserving parameterization. In the discrete case, we also guarantee the relation between the lower bound and the volume-/mass-preservation, and propose the spherical and ball volume-/mass-preserving parameterization algorithms. The numerical experiments indicate the accuracy and robustness of the proposed algorithms. The modified algorithms are applied to the manifold registration and deformation, showing the versatility of $n$-VSE.

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Zhong-Heng Tan, Tiexiang Li, Wen-Wei Lin, Shing-Tung Yau. 2024-02-01. $n$-Dimensional Volumetric Stretch Energy Minimization for Volume-/Mass-Preserving Parameterizations. https://arxiv.org/abs/2402.00380

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