Fast Disk Conformal Parameterization of Simply-connected Open Surfaces

TL;DR

This paper introduces a fast, stable, and bijective algorithm for conformally mapping simply-connected open surfaces onto the unit disk, improving efficiency and accuracy in surface parameterization tasks.

Contribution

It presents a novel, accelerated algorithm that enhances conformality, stability, and guarantees bijectivity in disk conformal parameterization of surfaces.

Findings

Significantly faster computation compared to existing methods.

Improved conformality and stability in the parameterization.

Effective correction of conformality distortions using iterative quasi-conformal schemes.

Abstract

Surface parameterizations have been widely used in computer graphics and geometry processing. In particular, as simply-connected open surfaces are conformally equivalent to the unit disk, it is desirable to compute the disk conformal parameterizations of the surfaces. In this paper, we propose a novel algorithm for the conformal parameterization of a simply-connected open surface onto the unit disk, which significantly speeds up the computation, enhances the conformality and stability, and guarantees the bijectivity. The conformality distortions at the inner region and on the boundary are corrected by two steps, with the aid of an iterative scheme using quasi-conformal theories. Experimental results demonstrate the effectiveness of our proposed method.