Tight Frames of Rational Splines and Application to CAD/CAM and Computer Graphics

基数样条（即具有等间距结的样条）为展示多分辨率分析（MRA）的数学结构提供了一个典型的例子，它是构建正交、半正交和双正交小波的一个非常强大的工具。然而，在实际应用中，通常需要具有任意结点的样条；尽管样条子空间的多层次（ML）结构仍然有效，但研究基数样条及其相应小波的傅里叶方法不再适用于这种更一般的研究。此外，即使对于基数设置，当样条-小波分析和样条-小波合成都需要紧凑支持（或有限时间持续时间）时，正交和半正交样条也必须用紧仿射（或小波）帧代替。此外，对于CAD/CAM应用，还需要有理b样条（或NURBS）来精确表示各种几何对象，如圆锥截面。这个建议是基于我们最近在基数样条紧框架上的结果和我们新开发的NURBS的完整ML结构的权重。我们建议引入非平稳时域技术，以建立具有任意结的紧支撑样条紧仿射框架的统一数学基础，更一般地说，是NURBS。随后将发展与这些新的基本功能有关的有效算法和计算方案，这些将用作CAD/CAM和计算机图形应用的分析和综合工具。我们还建议开发这些分析和设计工具的软件库，这些软件库将完全兼容工业标准，如IGES， STEP和PHIGS，用于CAD/CAM/CAE行业。

Cardinal splines (i.e. splines with equally-spaced knots) provide a canonical example for demonstrating the mathematical structure of multiresolution analysis (MRA), which is a very powerful tool for the construction of orthogonal, semi-orthogonal, and bi-orthogonal wavelets. However, for practical applications, splines with arbitrary knots are often needed; and although the multi-level (ML) structure of spline subspaces is still valid, the Fourier approach to the study of cardinal splines and their corresponding wavelets no longer applies to this more general study. In addition, even for the cardinal setting, orthogonal and semi-orthogonal splines must be replaced by tight affine (or wavelet) frames when compact support (or finite time-duration) is needed for both spline-wavelet analysis and spline-wavelet synthesis. Furthermore, for CAD/CAM applications, there is also the need of rational B-splines (or NURBS) for precise representation of various geometric objects such as conic sections.This proposal is based on our recent results on tight frames of cardinal splines and on our newly developed complete ML structure of NURBS in terms of their weights. We propose to introduce non-stationary time-domain techniques to build a unified mathematical foundation of tight affine frames of compactly supported splines with arbitrary knots, and more generally, of NURBS. This will be followed by development of efficient algorithms and computational schemes associated with these new basic functions, which will be used as analysis and synthesis tools for CAD/CAM and computer graphics applications. We also propose to develop software libraries of these analysis and design tools that will be fully compatible with the industrial standards such as IGES, STEP, and PHIGS, for the CAD/CAM/CAE industries.