Conformal Parameterization for High Genus Surfaces

Conformal mapping which preserves angle plays a fundamental role in computer graphics, geometry processing and parameterization. For discrete harmonic mapping has rigorous theoretical foundation and can be computed conveniently, we propose a new method to compute conformal mappings from high genus surfaces to parameterization domain based on the nonlinear diffusion of harmonic mapping. Given a high genus surface, we firstly establish an initial harmonic map from the fundamental domain of given surface to parameterization domain by using greedy algorithm and solving a sparse linear system. Then we reduce the harmonic energy by adjusting the boundary condition with nonlinear diffusion process. Finally we obtain a conformal parameterization by minimizing the harmonic energy. Experimental results show that the method is intrinsic and stable, and the final mapping preserves angles of surface triangular mesh well. This algorithm is robust, for it doesn’t require high quality meshes. And it has better angle preserving result comparing to other classical conformal mapping ideas. This method can be applied to many fields like parameterization, texture mapping and surface registration.