改进的全局参数化方法

针对多亏格曲面参数化变形较大、运算复杂度高的问题，提出一种改进的基于全纯1-形式的全局参数化方法。该方法以参数化的梯度场为出发点，采用更快速的同调群和上同调群计算方法。首先，利用简化的割图法计算曲面的同调群以确定其拓扑结构；其次，定义特定的调和函数计算闭合1-形式来构造由梯度场形成的线性空间的上同调群；然后，最小化调和能量将上同调群扩散为调和1-形式；最后，线性组合调和1-形式构造出全纯1-形式并在基本域上积分即得到参数化。由上同调群、同调群相关理论分析表明，该方法所得参数化是一种全局的、边界自由的共形映射。基于多组高亏格模型的实验证明，与原有基于全纯1-形式的全局参数化算法相比，本算法视觉效果更好，平均误差更小，运算效率更高。

Improved global parameterization method

Focusing on the issue that non-zero genus surface parameterization has large deformation and high computational complexity, an improved global parameterization approach based on holomorphic 1-form was proposed, which starts from the gradient field and adapts easier and faster method to compute homology and cohomology group. Firstly, a simplified cut graph method was used to construct homology group to determine the topology. Secondly, cohomology group of the linear space formed by the gradient field was calculated by defining special harmonic function to figure out closed 1-form. Thirdly, homology group was diffused to harmonic 1-form through minimizing harmonic energy. Finally, holomorphic 1-form was computed by combining linearly harmonic 1-form and the parameterization was obtained by integrating holomorphic 1-form on the surface basic domain. Theoretical analysis of homology group and cohomology group shows that the parameterization is a global, border-free conformal mapping. Experimental results based on non-zero genus model show that, compared with the former global parameterization based on holomorphic 1-form, the proposed algorithm has better visual effect, smaller average error and higher operation efficiency.