$n$-Dimensional Volumetric Stretch Energy Minimization for Volume-/Mass-Preserving Parameterizations
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914662647332864 |
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| author | Tan, Zhong-Heng Li, Tiexiang Lin, Wen-Wei Yau, Shing-Tung |
| author_facet | Tan, Zhong-Heng Li, Tiexiang Lin, Wen-Wei Yau, Shing-Tung |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_00380 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $n$-Dimensional Volumetric Stretch Energy Minimization for Volume-/Mass-Preserving Parameterizations Tan, Zhong-Heng Li, Tiexiang Lin, Wen-Wei Yau, Shing-Tung Numerical Analysis 49Q10, 52C26, 65D18, 65F05, 68U05 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. |
| title | $n$-Dimensional Volumetric Stretch Energy Minimization for Volume-/Mass-Preserving Parameterizations |
| topic | Numerical Analysis 49Q10, 52C26, 65D18, 65F05, 68U05 |
| url | https://arxiv.org/abs/2402.00380 |