The Theory of Computational Quasi-conformal Geometry on Point Clouds

TL;DR

This paper develops a new computational quasi-conformal geometry framework specifically for point cloud data, enabling shape analysis and deformation measurement without connectivity information.

Contribution

It introduces the concept of PC quasi-conformal maps and Beltrami coefficients, extending QC theory to point clouds and proving their convergence and effectiveness.

Findings

PC Beltrami coefficients measure local geometric distortions accurately.

Theoretical convergence of PCBC to continuous Beltrami coefficients as point cloud density increases.

Numerical validation confirms the effectiveness of PCBC in shape analysis.

Abstract

Quasi-conformal (QC) theory is an important topic in complex analysis, which studies geometric patterns of deformations between shapes. Recently, computational QC geometry has been developed and has made significant contributions to medical imaging, computer graphics and computer vision. Existing computational QC theories and algorithms have been built on triangulation structures. In practical situations, many 3D acquisition techniques often produce 3D point cloud (PC) data of the object, which does not contain connectivity information. It calls for a need to develop computational QC theories on PCs. In this paper, we introduce the concept of computational QC geometry on PCs. We define PC quasi-conformal (PCQC) maps and their associated PC Beltrami coefficients (PCBCs). The PCBC is analogous to the Beltrami differential in the continuous setting. Theoretically, we show that the PCBC converges to its continuous counterpart as the density of the PC tends to zero. We also theoretically and numerically validate the ability of PCBCs to measure local geometric distortions of PC deformations. With these concepts, many existing QC based algorithms for geometry processing and shape analysis can be easily extended to PC data.