Fast ellipsoidal conformal and quasi-conformal parameterization of genus-0 closed surfaces

TL;DR

This work proposes a new framework for computing ellipsoidal conformal and quasi-conformal parameterizations of genus-0 closed surfaces, in which the target parameter domain is anEllipsoid instead of a sphere.

Abstract

Surface parameterization plays a fundamental role in many science and engineering problems. In particular, as genus-0 closed surfaces are topologically equivalent to a sphere, many spherical parameterization methods have been developed over the past few decades. However, in practice, mapping a genus-0 closed surface onto a sphere may result in a large distortion due to their geometric difference. In this work, we propose a new framework for computing ellipsoidal conformal and quasi-conformal parameterizations of genus-0 closed surfaces, in which the target parameter domain is an ellipsoid instead of a sphere. By combining simple conformal transformations with different types of quasi-conformal mappings, we can easily achieve a large variety of ellipsoidal parameterizations with their bijectivity guaranteed by quasi-conformal theory. Numerical experiments are presented to demonstrate the effectiveness of the proposed framework.

Figures (10)

• Figure 1: Ellipsoidal conformal parameterizations obtained by our proposed framework for different genus-0 closed surfaces.
• Figure 2: An illustration of the stereographic projection ${P^N}:\mathbb{S}^2 \to \overline{\mathbb{C}}$.
• 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 orientation change are all related to the Beltrami coefficient $\mu_f$.
• Figure 4: Fast ellipsoidal conformal map (FECM) of genus-0 closed surfaces.
• Figure 5: Fast ellipsoidal quasi-conformal map (FEQCM) of genus-0 closed surfaces.