基于混合网格向量值细分曲面可计算光滑性研究

Subdivision surface is an important technical means for quickly and effectively drawing and displaying three dimensional forms, modellings with the aid of computer in Graphics. Recently, the study and consruction of quad/triangle subdivision schemes have attracted attention. The quad/triangle subdivision strats with a control net consisting of both quads and triangles, and produces finer and finer meshes with quads and triangles. The use of the quad/triangle structure for surface design is motivated by the fact that in CAD modelling, the designers often want to model certain regions with quad meshes and others with triangle meshes to get beeter vivid visul quality of subdivision surface with variety of topological geometry. This project will explore how to introduce the vector-valued feature into the hybrid mesh subdivision method, and study the vector-valued subdivision surface based on quad/triangle grid and its computable smoothness of the limit surface. The whole research process is arranged as following roughly. First, we will show that vector-valued quad/triangle subdivision scheme can be derived from an nonhomogeneous refinement equation which is famous in the field of harmonic analysis and wavelet analysis, and establish an one-to-one relation between them. Secondly, with the help of some results about nonhomogeneous refinement equations, such as the characterization of function spaces and related theory in the harmonic analysis, and joint spectral radius iterative algorithm in the numerical analysis, we will establish the smoothness theory of limit surface and correspodning estiment implementing algorithm. Finally, using this smooth depict theory, we will design some new vector-valued quad/triangle subdivision schemes with high smoothness, and apply these examples in the practical 3D form design. This research work is expected to solve the problem that existing quad/triangle subdivision scheme can not possess two properties of low algorithm complexity and high smoothness of limit surface at the same time. Moreover, the characterization of computable smoothness will also have an important theoretical significance to further research of vector-valued quad/triangle subdivision schemes.

计算机辅助几何设计中的细分曲面方法是快速、高效地设计三维形体绘制和展示的重要技术手段。基于混合网格细分曲面方法可以实现多种拓扑几何特征三维形体逼真表示，但仍未能解决在降低算法复杂性同时提高极限曲面光滑性的问题。本项目拟将向量值特征引入混合网格细分方法中，提出一种基于混合网格向量值细分曲面方法。并在分析加细模型特点的基础上，建立模型与小波分析领域中非齐次向量值加细方程的对应关系，借助调和分析中函数空间刻画的相关理论，结合数值分析中联合谱半径迭代算法，探寻并建立其极限曲面光滑性刻画理论以及相应的实现算法。本研究工作有望解决已有基于混合网格细分方法的局限性，提供一种低复杂度、高光滑性的三维形体绘制方法，并且可计算光滑性刻画也将为深入研究小波函数及曲面多尺度分析等应用奠定坚实的理论基础。

基于混合网格的细分曲面技术是计算机辅助几何设计中三维空间图形设计和修改曲面的一种有效的方法。针对已有混合网格细分方法不能同时兼顾算法复杂性和极限曲面光滑性的研究现状，本项目将向量值细分结构的特点引入混合网格细分方法中，研究基于混合网格向量值细分结构方法，并且在分析其加细模型特点的基础上，将对该细分结构的研究转化为调和分析和小波分析领域中非齐次加细方程的研究，利用调和分析中函数空间刻画等相关理论建立基于混合网格向量值细分结构生成极限曲面的光滑性刻画及相应的算法实现，从而达到光滑性刻画的可计算性。可计算的特点将对实际设计中构造具有较高光滑性细分方法有着关键性的指导意义。在此基础上，本项目进一步研究和建立了具有正交和双正交插值型向量滤波器的完整参数化，并构造出了一些具有高光滑性的向量小波，实验证明，这些向量小波在实际图像压缩中都有很好的效果。这些都为进一步研究插值型向量值混合细分曲面方法的图形多尺度表示打下了坚实的基础。