Laplace–Beltrami Operator on Point Clouds Based on Anisotropic Voronoi Diagram |
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| Authors: | Hongxing Qin Yi Chen Yunhai Wang Xiaoyang Hong Kangkang Yin Hui Huang |
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| Affiliation: | 1. Chongqing Key Laboratory of Computational Intelligence, Chongqing University of Posts & Telecommunications, Chongqing, China;2. College of Computer Science and Technology, Chongqing University of Posts & Telecommunications, Chongqing, China;3. Shandong University, Jinan, China;4. Simon Fraser University, Canada;5. Shenzhen University, Shenzhen, China |
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| Abstract: | The symmetrizable and converged Laplace–Beltrami operator ( ) is an indispensable tool for spectral geometrical analysis of point clouds. The  , introduced by Liu et al. LPG12] is guaranteed to be symmetrizable, but its convergence degrades when it is applied to models with sharp features. In this paper, we propose a novel  , which is not only symmetrizable but also can handle the point‐sampled surface containing significant sharp features. By constructing the anisotropic Voronoi diagram in the local tangential space, the  can be well constructed for any given point. To compute the area of anisotropic Voronoi cell, we introduce an efficient approximation by projecting the cell to the local tangent plane and have proved its convergence. We present numerical experiments that clearly demonstrate the robustness and efficiency of the proposed  for point clouds that may contain noise, outliers, and non‐uniformities in thickness and spacing. Moreover, we can show that its spectrum is more accurate than the ones from existing  for scan points or surfaces with sharp features. |
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| Keywords: | point‐based methods methods and applications point‐based graphics modelling computational geometry Computational Geometry and Object Modeling → Geometric algorithm |
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) is an indispensable tool for spectral geometrical analysis of point clouds. The 
, introduced by Liu et al. LPG12] is guaranteed to be symmetrizable, but its convergence degrades when it is applied to models with sharp features. In this paper, we propose a novel 
, which is not only symmetrizable but also can handle the point‐sampled surface containing significant sharp features. By constructing the anisotropic Voronoi diagram in the local tangential space, the 
can be well constructed for any given point. To compute the area of anisotropic Voronoi cell, we introduce an efficient approximation by projecting the cell to the local tangent plane and have proved its convergence. We present numerical experiments that clearly demonstrate the robustness and efficiency of the proposed 
for point clouds that may contain noise, outliers, and non‐uniformities in thickness and spacing. Moreover, we can show that its spectrum is more accurate than the ones from existing 
for scan points or surfaces with sharp features.