自由曲面的最优参数化及重新参数化关键算法研究

Freeform surfaces play an increasingly important role in contemporary Computer Aided Design and architecture design. While the manufactures of mechanical parts, architecture buildings, cars and planes are mainly concerned with the final geometric shapes, geometric algorithms for most CAD applications are highly dependent on the surface parameterization. The existing surface modification and generation algorithms pay more attention to the shape of the final surface while the parameterization of the final surface is neglected, which influences the results and possibility of the following algorithms. To solve the above problems, we plan to study the reparameterization and optimal parameterization of free form surfaces and apply the results to applications such as rendering, tessellation, intersection, surface fitting and curve generation in this research project. We focus on the following three subjects: 1) the composite quadratic reparameterization of NURBS surfaces and its application in the optimization of the surface parameterization; 2) the NURBS surfaces with parameterization characters such as uniformity, orthogonality, conformality, area-preserve and their combinations; 3) the applications of the surface reparameterization and optimal parameterization in surface fitting and curves on surfaces. This research project is very meaningful for improving the results of relevant surface algorithms, promoting their efficiency and robustness, better understanding the intrinsic characters of free form surfaces.

自由曲面（NURBS曲面）在现代制造业和建筑业中扮演了越来越重要的角色。机械零件、建筑、汽车、轮船以及飞机的制造商们主要关心最后生成的产品形状，而面向大多数应用的几何算法很大程度上依赖于产品模型中曲面当前的参数化。现有的曲面修改及生成算法往往只考虑最后生成曲面的形状，而忽略其参数特性，直接影响了后续操作的可行性及结果质量。针对上述问题，本项目拟研究自由曲面的最优参数化及重新参数化，并将之应用到包括渲染、离散、求交、曲面拟合以及曲线生成等算法中，重点解决：1）自由曲面的分段二次重新参数化及其在曲面参数优化中的应用；2）具有均匀、正交、保角、保面积以及它们组合特性的自由曲面显式表示及其构造算法；3）最优参数化及重新参数化在曲面拟合、曲面上曲线生成等算法中的应用。本项目对于提升曲面处理相关算法的结果质量，改进其效率和鲁棒性，帮助我们更好的理解自由曲面的内在本质具有非常重要的意义。

由于对产品形状美观以及连续性要求的提高，自由曲面（NURBS曲面）在现代制造业和建筑业中得到了越来越广泛的应用。自由曲面参数特性的好坏直接影响相关算法的结果质量；同时曲面满足某些参数特性可以提高相关算法的效率和鲁棒性。本课题针对自由曲面参数化中的若干关键技术展开一系列研究，主要包括：（1）自由曲面的保角、保面积参数化优化算法及其应用；（2）满足参数特性的自由曲面；（3）基于拟共性映射的曲面匹配和注册。取得了一系列成果并申请了相关专利，形成了一整套理论和稳定高效的优化算法，开发出集成自由曲面参数优化算法的原型系统，完成了原定研究计划。其中项目进行过程中，陆续有一系列成果发表在Computer-Aided Design, Graphical Models, Journal of Computational and Applied Mathematics(JCAM) , ECCV等国内外重要学术刊物和会议上，文章得到了多名审稿人的肯定和好评，项目共计发表文章10篇，其中发表国际SCI期刊文章5篇（SCI影响因子全部大于1.0）；获得相关发明专利1项。科学意义主要体现在：将现有的离散共形几何研究结果引入到自由曲面最优参数化研究中，提出了利用数值方法进行自由曲面参数特性优化的一系列算法。