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本任务，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，已被用于单连通与多连通开曲面的参数化。最近，一种球面密度均衡映射方法被开发用于亏格为零的闭曲面的参数化。然而，对于具有极端几何形状的亏格为零的闭曲面，使用球面域进行参数化可能导致较大的几何畸变。在本工作中，我们提出了一种新颖的方法，用于计算亏格为零的闭曲面到椭球域的密度均衡映射。这使得我们能够实现椭球等面积参数化以及具有可控面积变化的椭球参数化。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，从而能够生成椭球密度均衡拟共形映射，以实现密度均衡与拟共形性之间的平衡。利用我们提出的方法，可以显著提升亏格为零闭曲面的表面重网格化性能。通过对多种亏格为零闭曲面的大量实验结果，展示了我们提出方法的有效性。