Abstract: Aiming at the low efficiency and serious mapping distortion of current mesh parameterization, a mesh parameterization method with limiting distortion was proposed. Firstly, the original mesh model was pre-processed. After inputting the original 3D mesh model, the Half-Edge data structure was used to reorganize the mesh and the corresponding seams were generated by cutting the mesh model. The Tutte mapping was constructed to map the 3D mesh to a 2D convex polygon domain, that is to construct the 2D mesh model. Then, the mesh parameterization calculation with limiting distortion was performed. The Tutte-mapped 2D mesh model was used as the initial data for limiting distortion calculation, and the distortion metric function relative to the original 3D model mesh was established. The minimum value points of the metric function were obtained, which form the mapped mesh coordinate set. The mapped mesh was used as the input mesh to limit the distortion mapping, and the iteration termination condition was set. The iteration was performed cyclically until the termination condition was satisfied, and the convergent optimal mesh coordinates were obtained. In calculating the mapping distortion, the Dirichlet energy function was used to measure the isometric mapping distortion, and the Most Isometric Parameterizations (MIPS) energy function was used for the conformal mapping distortion. The minimum of the mapping distortion energy function was solved by proxy function combining assembly-Newton method. Finally, this method was implemented and a prototype system was developed. In the prototype system, mesh parameterization experiments for limiting isometric distortion and limiting conformal distortion were designed respectively, statistics and comparisons of program execution time and distortion energy falling were performed, and the corresponding texture mapping effects were displayed. Experimental results show that the proposed method has high execution efficiency, fast falling speed of mapping distortion energy and stable quality of optimal value convergence. When texture mapping is performed, the texture is evenly colored, close laid and uniformly lined, which meets the practical application standards.

Key words: mesh parameterization, limiting distortion, isometric mapping, conformal mapping, energy function, optimal coordinate point

摘要： 针对当前网格参数化效率较低、映射失真较严重的问题，提出一种限制失真的网格参数化方法。首先，预处理原始网格模型。输入原3D网格模型，采用Half-Edge数据结构来重新组织网格并切割网格模型产生相应的切缝；构建Tutte映射把3D网格映射到一个2D凸多边形域，即构建2D网格模型。然后，进行限制失真的网格参数化计算。将Tutte映射后的2D网格模型作为限制失真计算的初始数据，建立相对于原3D模型网格的失真度量函数；求得该度量函数的最小值点，即为映射后的网格坐标集合；将映射后的网格作为限制失真映射的输入网格，设定迭代终止条件，循环迭代直至迭代结束，得到收敛的最优网格坐标；在计算映射失真度时，针对等距映射失真采用Dirichlet能量函数度量，针对共形映射失真采用尽可能等距（MIPS）能量函数度量；在求解映射失真度量函数的最小值点时采用代理函数法结合组合牛顿法的最优解方法。最终，实现了该方法并开发了一个原型系统。在原型系统中，分别设计了限制等距失真和限制共形失真的网格参数化实验，对程序执行时间和失真能量下降情况进行了统计和对比，提供了相应的纹理映射效果展示。实验数据表明，所提出的方法执行效率高、映射失真能量下降快，最优值收敛质量稳定；纹理映射时纹理着色均匀、布局紧致、线条均匀，符合实际应用的标准。

关键词: 网格参数化, 限制失真, 等距映射, 共形映射, 能量函数, 最优坐标点