Toroidal density-equalizing map for genus-one surfaces

TL;DR

This paper introduces a novel algorithm for creating density-equalizing maps on genus-one (toroidal) surfaces, enabling shape deformation and parameterization with topological holes, expanding the applicability of density-equalizing techniques.

Contribution

The work develops the first density-equalizing mapping method specifically for toroidal surfaces, allowing for shape control and effective genus-one surface parameterizations.

Findings

Successfully computes density-equalizing maps for toroidal surfaces.

Enables shape deformation by prescribing different population functions.

Provides a new approach for toroidal surface parameterizations.

Abstract

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.