Geometric understanding of point clouds using Laplace-Beltrami operator

In this paper, we propose a general framework for approximating differential operator directly on point clouds and use it for geometric understanding on them. The discrete approximation of differential operator on the underlying manifold represented by point clouds is based only on local approximation using nearest neighbors, which is simple, efficient and accurate. This allows us to extract the complete local geometry, solve partial differential equations and perform intrinsic calculations on surfaces. Since no mesh or parametrization is needed, our method can work with point clouds in any dimensions or co-dimensions or even with variable dimensions. The computation complexity scaled well with the number of points and the intrinsic dimensions (rather than the embedded dimensions). We use this method to define the Laplace-Beltrami (LB) operator on point clouds, which links local and global information together. With this operator, we propose a few key applications essential to geome...

• Type: Journal
• Year: 2012
• Where: CVPR
• Authors: Jian Liang, Rongjie Lai, Tsz Wai Wong, Hongkai Zhao
• Keywords: Computation Complexity, Computer Vision, Cvpr 2012, Discrete Approximation, Point Clouds