Ellipsoidal Density-Equalizing Map for Genus-0 Closed Surfaces

TL;DR

This paper introduces a novel ellipsoidal density-equalizing map method for genus-0 closed surfaces, reducing distortion and improving surface remeshing by combining density diffusion with quasi-conformal mapping.

Contribution

The authors develop an ellipsoidal domain mapping technique for genus-0 surfaces, enhancing area preservation and control over geometric distortion compared to spherical methods.

Findings

Achieves ellipsoidal area-preserving parameterizations.

Produces maps with controlled area change and reduced distortion.

Significantly improves surface remeshing quality for genus-0 surfaces.

Abstract

Surface parameterization is a fundamental task in geometry processing and plays an important role in many science and engineering applications. In recent years, the density-equalizing map, a shape deformation technique based on the physical principle of density diffusion, has been utilized for the parameterization of simply connected and multiply connected open surfaces. More recently, a spherical density-equalizing mapping method has been developed for the parameterization of genus-0 closed surfaces. However, for genus-0 closed surfaces with extreme geometry, using a spherical domain for the parameterization may induce large geometric distortion. In this work, we develop a novel method for computing density-equalizing maps of genus-0 closed surfaces onto an ellipsoidal domain. This allows us to achieve ellipsoidal area-preserving parameterizations and ellipsoidal parameterizations with controlled area change. We further propose an energy minimization approach that combines density-equalizing maps and quasi-conformal maps, which allows us to produce ellipsoidal density-equalizing quasi-conformal maps for achieving a balance between density-equalization and quasi-conformality. Using our proposed methods, we can significantly improve the performance of surface remeshing for genus-0 closed surfaces. Experimental results on a large variety of genus-0 closed surfaces are presented to demonstrate the effectiveness of our proposed methods.