Laplace-Beltrami Operator on Point Clouds Based on Anisotropic Voronoi Diagram: Laplace-Beltrami Operator

The symmetrizable and converged Laplace-Beltrami operator (M) is an indispensable tool for spectral geometrical analysis of point clouds. The M, introduced by Liu etal.[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 M, 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 M 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 M 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 M for scan points or surfaces with sharp features.