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

基数样条法(即具有等间距节点的样条法)提供了一个典型的例子来演示多分辨率分析(MRA)的数学结构，它是构造正交、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，经常需要具有任意节点的样条线；尽管样条子空间的多层(ML)结构仍然有效，但研究基数样条线及其对应的小波的傅立叶方法不再适用于这一更一般的研究。此外，即使对于基数设置，当样条子波分析和样条子波综合都需要紧支集(或有限持续时间)时，也必须用紧仿射(或小波)框架来代替正交和半正交样条曲线。此外，对于CAD/CAM应用，还需要使用有理B-Spline(或NURBS)来精确表示各种几何对象，例如二次曲线。这一建议是基于我们最近关于基数样条紧框架的结果和我们新开发的关于NURBS的完全ML结构的权重。我们建议引入非平稳的时间域技术来建立具有任意节点的紧支承样条曲线的紧仿射框架的统一的数学基础，更广泛地说，是NURBS的。随后将开发与这些新的基本函数相关的有效算法和计算方案，作为CAD/CAM和计算机图形学应用程序的分析和综合工具。我们还建议为CAD/CAM/CAE行业开发这些分析和设计工具的软件库，这些工具将完全兼容IGES、STEP和PHIGS等行业标准。

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.