Density-equalizing maps are a class of mapping methods in which the shape deformation is driven by prescribed density information. In recent years, they have been widely used for data visualization on planar domains and planar parameterization of open surfaces. However, the theory and computation of density-equalizing maps for closed surfaces are much less explored. In this work, we develop a novel method for computing spherical density-equalizing maps for genus-0 closed surfaces. Specifically, we first compute a conformal parameterization of the given genus-0 closed surface onto the unit sphere. Then, we perform density equalization on the spherical domain based on the given density information to achieve a spherical density-equalizing map. The bijectivity of the mapping is guaranteed using quasi-conformal theory. We further propose a method for incorporating the harmonic energy and landmark constraints into our formulation to achieve landmark-aligned spherical density-equalizing maps balancing different distortion measures. Using the proposed methods, a large variety of spherical parameterizations can be achieved. Applications to surface registration, remeshing, and data visualization are presented to demonstrate the effectiveness of our methods.

翻译：密度均衡映射是一类由预设密度信息驱动形状变形的映射方法。近年来，它们被广泛用于平面域的数据可视化以及开放曲面的平面参数化。然而，闭合曲面上的密度均衡映射的理论与计算尚未得到充分探索。在本研究中，我们提出了一种计算亏格-0闭合曲面球面密度均衡映射的新方法。具体而言，我们首先将给定的亏格-0闭合曲面共形参数化到单位球面上。随后，基于给定的密度信息在球面域上进行密度均衡处理，从而获得球面密度均衡映射。利用拟共形理论确保映射的双射性。此外，我们进一步提出将调和能量与地标约束纳入公式的方法，以实现平衡不同扭曲度量的地标对齐球面密度均衡映射。通过所提出的方法，可获得多种球面参数化结果。通过曲面配准、网格重剖分和数据可视化的应用实例，验证了本方法的有效性。