PCBC: Quasiconformality of Point Cloud Mappings

Abstract

The study of surface mappings or deformations plays an important role for various applications in computer visions and graphics. An accurate and effective method to measure and control geometric distortions of the mapping is therefore necessary. Quasiconformality, which captures the local geometric distortion of a surface mapping, is a useful tool for this purpose. In discrete setting, surfaces are often represented by point clouds. Although quasiconformal theories are well developed in the continuous setting, the concept of quasiconformality on point clouds is still lacking. In this paper, we propose a geometric quantity, called the Point Cloud Beltrami Coefficient (PCBC), to measure quasiconformality of a point cloud mapping. Its ability to measure the local geometric distortion of a mapping is theoretically and numerically validated. Moreover, we propose an algorithm to solve for a point cloud mapping with a prescribed PCBC. Numerical experiments have been carried out on both synthetic and real point cloud data, which demonstrate the efficacy of the proposed algorithm.