Global Conformal Parameterization via an Implementation of Holomorphic Quadratic Differentials

We propose an algorithm to compute global conformal parameterizations of high-genus meshes, which is based on an implementation of holomorphic quadratic differentials. First, we design a novel diffusion method which is capable of computing a pole-free discrete harmonic measured foliation. Second, we propose a definition for discrete holomorphic quadratic differential which consists of a horizontal and a vertical harmonic measured foliation. Third, we present a practical algorithm to approximate the discrete natural coordinates for a holomorphic quadratic differential, which represents a flat metric with cones conformal to the original metric, i.e., a parameterization. Finally, we apply the discrete natural coordinates for parameterization of high genus meshes. Our parameterization method is global conformal and simple to implement. The advantage of our method over the approach based on holomorphic differential one-forms is that ours has a larger space of parameterizations. We demonstrate our approach with hundreds of configurations on dozens of meshes to show its robustness on conformal parameterization.