Parallelizable global quasi-conformal parameterization of multiply-connected surfaces via partial welding

Abstract

Conformal and quasi-conformal mappings have widespread applications in imaging science, computer vision and computer graphics, such as surface registration, segmentation, remeshing, and texture map compression. While various conformal and quasi-conformal parameterization methods for simplyconnected surfaces have been proposed, efficient parameterization algorithms for multiply-connected surfaces are less explored. In this paper, we propose a novel parallelizable algorithm for computing the global conformal and quasi-conformal parameterization of multiply-connected surfaces onto a 2D circular domain using variants of the partial welding method and the Koebe's iteration. The main idea is to first partition a multiply-connected surface into several subdomains and compute the free-boundary conformal or quasi-conformal parameterizations of them respectively, and then apply a variant of the partial welding algorithm to reconstruct the global mapping. We apply the Koebe's iteration together with the geodesic algorithm to the boundary points and welding paths before and after the global welding to transform all the boundaries to circles conformally. After getting all the updated boundary conditions, we obtain the global parameterization of the multiply-connected surface by solving the Laplace equation for each subdomain. Using this divide-and-conquer approach, the global conformal and quasi-conformal parameterization of surfaces can be efficiently computed. Experimental results are presented to demonstrate the effectiveness of our proposed algorithm. More broadly, the proposed shift in perspective from solving a global quasi-conformal mapping problem to solving multiple local mapping problems paves a new way for computational quasi-conformal geometry.