Smoothing of Meshes and Point Clouds Using Weighted Geometry-Aware Bases

Recently, Sorkine et al. proposed a least squares based representation of meshes, which is suitable for compression and modeling. In this paper we look at this representation from the viewpoint of Tikhonov regularization. We show that this viewpoint yields a smoothing algorithm, which can be seen as an approximation of the shape using weighted geometry aware bases, where the weighting factor is determined by the algorithm. The algorithm combines the Laplacian smoothing approach with the smoothing spline approach, where a global deviation constraint is imposed on the approximation. We use the generalized Laplacian matrix to measure smoothness and show how it can be modified in order to obtain smoothing behavior similar to that of curvature flow and feature preserving smoothing algorithms. The method is applicable to meshes, polygonal curves and point clouds in arbitrary dimensional spaces.