ITS: Implicit Thin Shell for Polygonal Meshes
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arXiv
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| Auteurs principaux: | , , , , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912102877233152 |
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| author | Wen, Huibiao Wang, Lei Zhang, Yunxiao Chen, Shuangmin Xin, Shiqing Deng, Chongyang He, Ying Wang, Wenping Tu, Changhe |
| author_facet | Wen, Huibiao Wang, Lei Zhang, Yunxiao Chen, Shuangmin Xin, Shiqing Deng, Chongyang He, Ying Wang, Wenping Tu, Changhe |
| contents | In computer graphics, simplifying a polygonal mesh surface~$\mathcal{M}$ into a geometric proxy that maintains close conformity to~$\mathcal{M}$ is crucial, as it can significantly reduce computational demands in various applications. In this paper, we introduce the Implicit Thin Shell~(ITS), a concept designed to implicitly represent the sandwich-walled space surrounding~$\mathcal{M}$, defined as~$\{\textbf{x}\in\mathbb{R}^3|ε_1\leq f(\textbf{x}) \leq ε_2, ε_1< 0, ε_2>0\}$. Here, $f$ is an approximation of the signed distance function~(SDF) of~$\mathcal{M}$, and we aim to minimize the thickness~$ε_2-ε_1$. To achieve a balance between mathematical simplicity and expressive capability in~$f$, we employ a tri-variate tensor-product B-spline to represent~$f$. This representation is coupled with adaptive knot grids that adapt to the inherent shape variations of~$\mathcal{M}$, while restricting~$f$'s basis functions to the first degree. In this manner, the analytical form of~$f$ can be rapidly determined by solving a sparse linear system. Moreover, the process of identifying the extreme values of~$f$ among the infinitely many points on~$\mathcal{M}$ can be simplified to seeking extremes among a finite set of candidate points. By exhausting the candidate points, we find the extreme values~$ε_1<0$ and $ε_2>0$ that minimize the thickness. The constructed ITS is guaranteed to wrap~$\mathcal{M}$ rigorously, without any intersections between the bounding surfaces and~$\mathcal{M}$. ITS offers numerous potential applications thanks to its rigorousness, tightness, expressiveness, and computational efficiency. We demonstrate the efficacy of ITS in rapid inside-outside tests and in mesh simplification through the control of global error. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01488 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | ITS: Implicit Thin Shell for Polygonal Meshes Wen, Huibiao Wang, Lei Zhang, Yunxiao Chen, Shuangmin Xin, Shiqing Deng, Chongyang He, Ying Wang, Wenping Tu, Changhe Graphics In computer graphics, simplifying a polygonal mesh surface~$\mathcal{M}$ into a geometric proxy that maintains close conformity to~$\mathcal{M}$ is crucial, as it can significantly reduce computational demands in various applications. In this paper, we introduce the Implicit Thin Shell~(ITS), a concept designed to implicitly represent the sandwich-walled space surrounding~$\mathcal{M}$, defined as~$\{\textbf{x}\in\mathbb{R}^3|ε_1\leq f(\textbf{x}) \leq ε_2, ε_1< 0, ε_2>0\}$. Here, $f$ is an approximation of the signed distance function~(SDF) of~$\mathcal{M}$, and we aim to minimize the thickness~$ε_2-ε_1$. To achieve a balance between mathematical simplicity and expressive capability in~$f$, we employ a tri-variate tensor-product B-spline to represent~$f$. This representation is coupled with adaptive knot grids that adapt to the inherent shape variations of~$\mathcal{M}$, while restricting~$f$'s basis functions to the first degree. In this manner, the analytical form of~$f$ can be rapidly determined by solving a sparse linear system. Moreover, the process of identifying the extreme values of~$f$ among the infinitely many points on~$\mathcal{M}$ can be simplified to seeking extremes among a finite set of candidate points. By exhausting the candidate points, we find the extreme values~$ε_1<0$ and $ε_2>0$ that minimize the thickness. The constructed ITS is guaranteed to wrap~$\mathcal{M}$ rigorously, without any intersections between the bounding surfaces and~$\mathcal{M}$. ITS offers numerous potential applications thanks to its rigorousness, tightness, expressiveness, and computational efficiency. We demonstrate the efficacy of ITS in rapid inside-outside tests and in mesh simplification through the control of global error. |
| title | ITS: Implicit Thin Shell for Polygonal Meshes |
| topic | Graphics |
| url | https://arxiv.org/abs/2411.01488 |