Density-equalizing map is a shape deformation technique originally developed for cartogram creation and sociological data visualization on planar geographical maps. In recent years, there has been an increasing interest in developing density-equalizing mapping methods for surface and volumetric domains and applying them to various problems in geometry processing and imaging science. However, the existing surface density-equalizing mapping methods are only applicable to surfaces with relatively simple topologies but not surfaces with topological holes. In this work, we develop a novel algorithm for computing density-equalizing maps for toroidal surfaces. In particular, different shape deformation effects can be easily achieved by prescribing different population functions on the torus and performing diffusion-based deformations on a planar domain with periodic boundary conditions. Furthermore, the proposed toroidal density-equalizing mapping method naturally leads to an effective method for computing toroidal parameterizations of genus-one surfaces with controllable shape changes, with the toroidal area-preserving parameterization being a prime example. Experimental results are presented to demonstrate the effectiveness of our proposed methods.

翻译：密度均衡映射是一种形状变形技术，最初为平面地理地图上的面积变形图制作和社会学数据可视化而开发。近年来，开发适用于曲面和体积域的密度均衡映射方法并将其应用于几何处理与成像科学中的各类问题，引起了越来越多的关注。然而，现有的曲面密度均衡映射方法仅适用于拓扑结构相对简单的曲面，无法处理带有拓扑孔洞的曲面。在本工作中，我们提出了一种计算环面密度均衡映射的新算法。特别地，通过在环面上指定不同的人口分布函数，并在具有周期性边界条件的平面域上执行基于扩散的变形，可以轻松实现不同的形状变形效果。此外，所提出的环面密度均衡映射方法自然地引向一种有效计算亏格为一曲面的环面参数化的方法，该方法具有可控的形状变化，其中环面保面积参数化即为一典型示例。实验结果展示了我们提出的方法的有效性。