A Parallel Algorithm for Global Conformal Parameterization using Welding Maps

We propose an efficient parallel algorithm for computing the global conformal parameterization of a simply-connected open or closed surface embedded in R3. The method is specially designed for large-scale problems, of which surface meshes contain a large number of vertices. The main idea is based on the observation that a global conformal parameterization can be retrieved from local parameterizations of arbitrary patches using the welding maps. A numerical algorithm for the global conformal parameterization of a high-resolution surface mesh can then be developed by conformally glueing the local conformal parameterizations of various patches. Each local conformal parameterization can be computed in parallel. Hence, our proposed algorithm is parallelizable. The procedure conformal glueing can be efficiently carried out through a partial zipper algorithm, which involves only the boundary vertices. Extensive experiments on different simply-connected open and closed surface meshes demonstrate the efficacy of our proposed method.

Example 1:

Animation

Example 2:

Example 3: