Ellipsoidal Density-Equalizing Map for Genus-0 Closed Surfaces

TL;DR

This work develops a novel method for computing density-equalizing maps of genus-0 closed surfaces onto an ellipsoidal domain that allows to achieve ellipsoidal area-preserving parameterizations and ellipsoidal parameterizations with controlled area change and proposes an energy minimization approach that combines density-equalizing maps and quasi-conformal maps.

Abstract

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.

Figures (15)

• Figure 1: An illustration of the proposed ellipsoidal density-equalizing mapping method (EDEM) and ellipsoidal density-equalizing quasi-conformal mapping method (EDEQ). (a) Given any input genus-0 closed surface, we can apply the proposed EDEM method to compute ellipsoidal density-equalizing maps to ellipsoidal domains with different prescribed elliptic radii. (b) Given any input genus-0 closed surface (top left), we can start with an initial spherical parameterization (top right) and then apply the proposed EDEQ method to simultaneously optimize the elliptic radii of the domain and the mapping onto it, thereby achieving an ellipsoidal density-equalizing quasi-conformal map with minimal geometric distortions (bottom).
• Figure 2: An illustration of density-equalizing maps. The density flow enlarges the regions with high density and shrinks the regions with low density.
• Figure 3: An illustration of quasi-conformal maps. The Beltrami coefficient determines the conformality distortions
• Figure 4: Ellipsoidal density-equalizing maps of ellipsoidal surfaces. Each row shows one example. (a) An example with discontinuous input density. (b) An example with continuous input density. Left to right: The initial ellipsoidal surface color-coded with the initial density, the final EDEM result color-coded with the initial density, the histogram of the initial density, and the histogram of the final density.
• Figure 5: Additional examples of ellipsoidal density-equalizing maps of ellipsoidal surfaces. For each example, the left figure shows the input surface color-coded with the initial density, and the right figure shows the EDEM result color-coded with the initial density. (a) An ellipsoid with non-uniformly distributed mesh elements and a prescribed discontinuous density. (b) An elongated ellipsoid with a prescribed continuous density. (c) An ellipsoid with non-uniformly distributed mesh elements and a complex prescribed density. (d) A sphere with a prescribed continuous density.