洪程　章登义　苏科华　武小平　郑昌金

摘要：

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。

关键词：

全局参数化；全纯1形式；调和能量；共形映射；割图

中图分类号：

TP391.41

文献标志码：A

Abstract：

Focusing on the issue that nonzero genus surface parameterization has large deformation and high computational complexity， an improved global parameterization approach based on holomorphic 1form 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 1form. Thirdly， homology group was diffused to harmonic 1form through minimizing harmonic energy. Finally， holomorphic 1form was computed by combining linearly harmonic 1form and the parameterization was obtained by integrating holomorphic 1form on the surface basic domain. Theoretical analysis of homology group and cohomology group shows that the parameterization is a global， borderfree conformal mapping. Experimental results based on nonzero genus model show that， compared with the former global parameterization based on holomorphic 1form， the proposed algorithm has better visual effect， smaller average error and higher operation efficiency.

英文关键词Key words：

global parameterization； holomorphic 1form； harmonic energy； conformal mapping； cut graph

0引言

曲面参数化是一种几何处理的基本工具，它将三维曲面投射到标准域（如平面区域、球面、多面体），从而把复杂几何模型的几何处理操作转移到简单几何模型上，大大简化数字几何处理操作，因此广泛应用于纹理映射、曲面可视化、重新网格化和曲面拟合等过程。

参数化本质是一种三维网格曲面与参数域之间的可逆映射关系，一个有效的参数化必须是双射且不能存在网格重叠。理想情况是三角网格曲面与参数域之间的映射等距，但除可扩展曲面外绝大多数曲面难以实现，因此在该过程中必定产生图形扭曲。……