Spherical Density-Equalizing Map for Genus-0 Closed Surfaces

TL;DR

We address the problem of constructing bijective density-equalizing maps for genus-0 closed surfaces by directly parameterizing onto the unit sphere and performing diffusion-based density equalization on the spherical domain, with bijectivity guaranteed via quasi-conformal theory. The core methods, SDEM and LSDEM, combine diffusion-driven deformation with conformality and landmark constraints through a combined energy and an overlap-correction scheme, enabling both area-preserving and landmark-aligned spherical parameterizations. The approach yields accurate surface registrations, high-quality remeshing, and intuitive spherical data visualization, demonstrated on a range of genus-0 models and supported by quantitative metrics (e.g., density variance reduction, zero overlaps, low Beltrami distortion). The work advances density-equalizing mapping to closed-surface topology, providing practical tools for registration, remeshing, and visualization with strict bijectivity. It also outlines future extensions to higher-genus surfaces and non-triangle representations.

Abstract

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.

Figures (17)

• Figure 1: Examples of the spherical density-equalizing maps obtained by our SDEM method. (Left) The David model and two spherical density-equalizing maps obtained by our SDEM method, with the nose region enlarged or shrunk based on the prescribed population. (Right) The Chinese Lion model and the spherical area-preserving parameterization obtained by our SDEM method.
• Figure 2: An illustration of density-equalizing maps. During the diffusion process, the regions with high density will be enlarged and the regions with low density will be shrunk.
• Figure 3: An illustration of quasi-conformal maps. Under a quasi-conformal map $f$, an infinitesimal circle is mapped to an infinitesimal ellipse with bounded eccentricity. The maximal magnification, maximal shrinkage, and maximal dilation are all related to the Beltrami coefficient $\mu$.
• Figure 4: Density-equalizing map on a sphere. (a) A spherical surface color-coded with a prescribed density $\rho$. The diffusion of $\rho$ induces a density flux, thereby giving a velocity field on the sphere as indicated by the green arrows. (b) Under the spherical density-equalizing map, the region with a higher density (in yellow) expands and the region with a lower density (in purple) shrinks.
• Figure 5: Bijective spherical density-equalizing map (SDEM)

Theorems & Definitions (2)

• proof
• proof