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.

中文翻译：

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通过全纯二次微分实现的全局保形参数化

我们提出了一种算法来计算高属网格的全局共形参数化，该算法基于全纯二次微分的实现。首先，我们设计了一种新颖的扩散方法，该方法能够计算无极离散谐波测量叶面。其次，我们提出了一个定义由水平和垂直组成的离散全纯二次微分谐波测量叶面。第三，我们提出了一种实用的算法来近似全纯二次微分的离散自然坐标，它表示具有与原始度量共形的锥体的平面度量，即参数化。最后，我们应用用于高属网格参数化的离散自然坐标。我们的参数化方法是全局保形且易于实现。我们的方法相对于基于全纯微分一式的方法的优势在于我们的方法具有更大的参数化空间。我们在数十个网格上使用数百种配置展示了我们的方法，以展示其对保形参数化的鲁棒性。