A Linear Formulation for Disk Conformal Parameterization of Simply-Connected Open Surfaces

TL;DR

This paper introduces an efficient algorithm for disk conformal parameterization of simply-connected open surfaces using a double covering technique, Möbius transformation, and quasi-conformal mappings, significantly improving computational speed while maintaining accuracy.

Contribution

The novel approach combines double covering, Möbius transformation, and quasi-conformal mappings to achieve fast, accurate, and bijective disk conformal parameterization of open surfaces.

Findings

Over 60% reduction in computational time.

Maintains comparable accuracy and bijectivity.

Effective in texture mapping applications.

Abstract

Surface parameterization is widely used in computer graphics and geometry processing. It simplifies challenging tasks such as surface registrations, morphing, remeshing and texture mapping. In this paper, we present an efficient algorithm for computing the disk conformal parameterization of simply-connected open surfaces. A double covering technique is used to turn a simply-connected open surface into a genus-0 closed surface, and then a fast algorithm for parameterization of genus-0 closed surfaces can be applied. The symmetry of the double covered surface preserves the efficiency of the computation. A planar parameterization can then be obtained with the aid of a M\"obius transformation and the stereographic projection. After that, a normalization step is applied to guarantee the circular boundary. Finally, we achieve a bijective disk conformal parameterization by a composition of quasi-conformal mappings. Experimental results demonstrate a significant improvement in the computational time by over 60%. At the same time, our proposed method retains comparable accuracy, bijectivity and robustness when compared with the state-of-the-art approaches. Applications to texture mapping are presented for illustrating the effectiveness of our proposed algorithm.