球面样条以及基于球面参数化下纹理映射的研究

Spline is an excellent tool for numerical approximation, which is a perfect result of the combination of approximation theory and computer theory. In modern computational work, spline is the major tool for engineers and CAD software since it is easy to represent and efficient to evaluate. The spline finite element is also an efficient tool in finite element analysis. ..The theory of multivariate splines began its rapid development in the early nineties, resulting in several thousand research papers and a number of books. This development was largely over by 1990, and the bulk of what is known today was treated already in the classic monographs of deBoor and Schumaker. Multivariate splines have become an essential tool in a wide variety of application areas, and are by now a standard topic in numerical analysis books. Prior to 2000 there were some results for using spherical piecewise polynomials in the finite element method, but spherical splines had attracted relatively little attention. ..By researching the properties of spherical spline, we know that because of the homogeneous of the spherical spline, it has a disadvantage at the approximation accuracy. The research content of this subject has two aspects: 1) Theoretical research. How to modify the spherical spline method in order to improve its approximation order. The main feature of spherical spline is that its defined on a standard sphere. So the spherical spline method is especially suitable for dealing with the data defined on the sphere. 2)Application research. How to construct a spherical spline in order to get a new algorithm of texture mapping by using spherical spline function as texture mapping function. On the other hand, the research objective of this subject has three aspects: 1) Present a new method in constructing a spherical spline to improve the approximation order. 2) Design a new algorithm in texture mapping by using spherical spline. 3) Finish a matlab program which can be used in texture mapping. Convert all the algorithms into computer program in order to solve the texture mapping base on spherical parameterization.

样条函数以其便于计算机存储、计算稳定、局部支集以及便于交互控制等优点，成为各类工程计算、计算机辅助制造/设计和几何建模等大型软件的重要数学工具之一。由于球面样条具有奇次性，只能重构相同奇偶性的函数，因此在计算的逼近精度上有所不足。本课题主要研究内容分为两个方面：1）理论研究。如何改进球面样条函数方法，提高其逼近精度。球面样条的最大特点是定义域为一标准球面，所以特别适合于处理类似球体上的数据问题（如球面参数化后的数据）。2）应用研究。利用球面样条函数作为映射函数，进一步改进球面样条函数方法，针对球面样条提出对应的纹理映射计算算法，同时完善该算法的计算机实现以及相关的理论问题分析。最后得到一份可以在计算机上运行的算法程序，用于解决球面参数化下纹理映射的问题，为数值逼近和图形设计领域作出贡献。

样条函数以其便于计算机存储、计算稳定、局部支集以及便于交互控制等优点，成为各类工程计算、计算机辅助制造/设计和几何建模等大型软件的重要数学工具之一。本项目主要分为球面样条函数理论研究和球面样条应用研究两个部分，具体的研究成果可以分为以下几个方面：.（1）.解决了球面样条在逼近精度上不足的问题；.（2）.解决了利用球面样条函数来进行纹理映射的问题；.（3）.解决了设置球面样条函数内部单元的约束矩阵来调整纹理映射效果的问题。