Ellipsoidal Density-Equalizing Map for Genus-0 Closed Surfaces (2410.12331v1)

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.

• The paper presents ellipsoidal density-equalizing mapping methods (EDEM and EDEQ) that significantly reduce distortion compared to traditional spherical approaches.
• It employs an iterative conformal map followed by numerical adjustments using density diffusion and overlap correction to ensure bijectivity.
• Experimental results show enhanced area preservation and improved conformal properties, offering robust solutions for applications like medical visualization and surface remeshing.

Overview of Ellipsoidal Density-Equalizing Maps for Genus-0 Surfaces

This paper introduces a method for ellipsoidal density-equalizing mapping (EDEM) of genus-0 closed surfaces, addressing the limitations of spherical parameterization and proposing an advanced energy minimization approach to balance density equalization with quasi-conformal mapping.

Surface parameterization is integral to geometry processing, facilitating tasks like surface registration and remeshing. Traditional methods often map genus-0 surfaces onto spheres, but this leads to significant distortion for elongated or otherwise extreme geometries. The authors propose mapping these surfaces onto ellipsoids, which better accommodate their shapes and mitigate distortion.

Methodology

The paper introduces two mapping techniques: EDEM and EDEQ (ellipsoidal density-equalizing quasi-conformal maps). EDEM focuses on achieving a uniform density distribution by adjusting a surface's mapping based on a prescribed density function using the principle of density diffusion. This is particularly advantageous for area-preserving parameterizations and those requiring controlled area changes.

The EDEM process involves an initial conformal map onto an ellipsoid followed by iterative adjustments through a numerical solution of the density diffusion equation. The method further ensures mapping bijectivity with an overlap correction scheme.

In parallel, the EDEQ method incorporates an optimization of the ellipsoid's shape alongside the mapping process to achieve a balance between area and angle preservation. This dual optimization considers both the elliptic radii of the mapping domain and the minimization of a combined energy function involving density-equalizing and quasi-conformal terms.

Experimental Results

Experiments on a variety of ellipsoidal and genus-0 closed surfaces demonstrate the effectiveness of EDEM and EDEQ. The methods significantly reduce area distortion when compared to traditional spherical methods and improve the distribution of triangle elements, an essential factor in surface remeshing. Results indicate that EDEQ consistently improves conformal properties, offering a robust solution for complex surface parameterizations.

Implications and Future Work

The methodology presents a significant advancement for parameterizing genus-0 surfaces with complex geometries, reducing distortion more effectively than spherical methods. Practically, this enhances applications in areas like medical visualization and shape modeling, where maintaining area and shape conformity is crucial.

The authors suggest future directions including the integration of landmark-matching capabilities and curvature-based criteria, which would expand the applicability of the method to areas requiring precise mappings, such as biomedical imaging and detailed geographic modeling.

In conclusion, this paper provides a substantive contribution to geometric processing, introducing a novel approach to mapping genus-0 closed surfaces that balances area preservation and angle distortion amidst challenging geometries. The proposed method has the potential to significantly impact various fields requiring precise surface manipulations.