Laplace-Beltrami Operator on Point Clouds Based on Anisotropic Voronoi Diagram
Laplace-Beltrami Operator on Point Clouds Based on Anisotropic Voronoi Diagram
复制标题
基于各向异性Voronoi图的点云Laplace-Beltrami算子
DOI：
10.1111/cgf.13315
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发表时间：
2018
影响因子：
2.5
通讯作者：
Huang Hui
中科院分区：
文献类型：
--
作者：
Qin Hongxing;Chen Yi;Wang Yunhai;Hong Xiaoyang;Yin Kangkang;Huang Hui
可对称化和收敛的Laplace-Beltrami算子（）是点云谱几何分析不可缺少的工具。由Liu等人引入的[LPG 12]保证是可对称的，但当其应用于具有尖锐特征的模型时，其收敛性会降低。在本文中，我们提出了一种新的，这不仅是对称化，但也可以处理点采样表面包含显着的尖锐特征。通过构造局部切空间中的各向异性Voronoi图，可以很好地构造出任意给定点的Voronoi图。为了计算各向异性Voronoi胞的面积，我们引入了一种有效的近似方法，将胞投影到局部切平面上，并证明了其收敛性。我们提出的数值实验清楚地表明了所提出的点云的鲁棒性和效率，这些点云可能包含噪声，离群值以及厚度和间距的不均匀性。此外，我们可以证明，它的频谱是更准确的比现有的扫描点或表面的尖锐特征。
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.