| Point cloud meshing, sometimes called surface reconstruction or reverse engineering, is a procedure that takes a set of points sampled from a 3D surface, and reconstructs a triangle mesh that closely approximates the surface from which it was sampled, both in terms of its geometry and its topology. Point cloud meshing is increasingly important in geometric modeling for generating surfaces from data points captured from real objects, often by laser range scanners but also by hand-held digitizers, computer vision techniques, or other technologies.; We present an algorithm to mesh point clouds sampled from a closed manifold surface of genus one and higher. Our method relies on a global parameterization of the point cloud to the plane, so no segmentation of the point cloud is required. In order to parameterize the point cloud we compute two discrete one-forms. For point clouds of genus g greater than one we provide an algorithm to select a pair of oneforms from the 2g linear basis of harmonic one-forms that minimize the distortion caused by the "doubly wheel vertices" and/or "doubly convex faces" that arise in the parameterization as a consequence of the Poincare Hopf Index theorem. Our algorithm identifies and addresses the "doubly wheel vertices" and/or "doubly convex faces" in the parameterization, in order to produce an output that minimizes the distortion caused by their presence.; When some mild conditions on sampling density are satisfied, the algorithm generates a closed mesh which interpolates the input and is geometrically similar to the input mesh. We demonstrate the success of our method on a variety of models with different genera and compare our results with the Power crust, another point cloud reconstruction algorithm. In the genus one case our algorithm generates a closed toroidal manifold. In the higher genus cases the quality is comparable to the quality of power crust. |
