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.

翻译：曲面参数化是几何处理中的基础任务，在众多科学与工程应用中发挥着重要作用。近年来，基于密度扩散物理原理的形状变形技术——密度均衡映射，已被用于单连通和多连通开曲面的参数化。最近，针对亏格0闭曲面的参数化，研究人员开发了球面密度均衡映射方法。然而，对于具有极端几何形状的亏格0闭曲面，使用球面域进行参数化可能产生较大的几何畸变。本文提出了一种计算亏格0闭曲面到椭球体域的密度均衡映射的新方法。该方法能够实现椭球体面积保持参数化以及可控面积变化的椭球体参数化。我们进一步提出了一种结合密度均衡映射与拟共形映射的能量最小化方法，从而生成椭球体密度均衡拟共形映射，以在密度均衡与拟共形性之间取得平衡。利用所提方法，我们能够显著提升亏格0闭曲面的曲面重网格化性能。在多种亏格0闭曲面上的实验结果展示了所提方法的有效性。