Mathematical Sciences: A Unified Approach to Discrete Data Representation and Wavelet Analysis

在这个项目中涉及到离散数据表示和小波分析的四个问题。第一个领域是建立一种仅基于子空间嵌套序列而不基于任何双尺度关系来构造小波的统一方案。第二个领域是研究多元小波和具有矩阵扩张的框架。第三个领域是用径向基函数（RBF's）表示离散数据。第四个领域是基于样条技术的自适应小波的研究。特别感兴趣的是用样条和径向基函数建模，以及构造和分析相应的小波。鉴于大多数建模函数，包括所有有趣的标准径向基函数，都不满足双尺度（或精化）关系，提出了一种不依赖于这种关系的小波构造的统一方案。这是拟议研究的第一个领域。第二个领域以多元小波为中心，主要关注的是矩阵膨胀。例如，对于均值为零的有限平方可积函数族，将研究仿射算子和相应的Littlewood-Paley能量函数的有界性，并建立保证下界的充分条件。第三个领域集中于局部余弦Riesz基及其对偶的研究。对偶或双正交基必须提高计算效率。由于这个和其他原因，样条技术，如结插入和结去除，将进行研究。最后一个领域涉及离散数据的建模。对于高维，径向基函数似乎是最有效的工具，重点将放在涉及这些函数的各种线性和非线性问题上。这个研究方案的中心主题是开发数学工具，用于在欧几里得空间或封闭流形（如单位球）中分析和综合一维或高维离散数据。本研究提出了一种统一的方法，以一种有效的方式表示任何离散数据集，以便“小波”分析可以很容易地进行。样条函数和径向基函数都将用于数据表示。众所周知，样条函数为数据建模提供了一个非常强大的工具，因为它们具有最理想的特性，如自适应实现的灵活性、计算效率和本地化能力。另一方面，径向基函数在处理高维数据集时更强大，特别是当数据信息是随机获取的时候。我们将构建小波，并利用样条函数和径向基函数开发小波算法。此外，为了分析形状变化的非平稳数据，研究人员将研究通过使用样条或径向基函数作为包络（或窗口）来使用局部余弦变换。自适应算法也将被开发。小波被认为是一种非常强大的数据分析工具的主要原因之一是它们的快速算法。为了有效实现和快速计算，本研究将重点研究如何同时使用样条函数和径向基函数构造小波。软件开发将同时进行，主要目标之一是确保算法也可以在硬件中实现。硬件原型开发中的工业标准也将被牢记。

9505460 Chui Four problem areas concerning discrete data representation and wavelet analysis are addressed in this project. The first area is to establish a unified scheme for constructing wavelets based only on nested sequences of subspaces but not on any two-scale relation. The second area is the study of multivariate wavelets and frames with matrix dilations. The third area is on representation of discrete data by radial basis functions (RBF's). The fourth area is the investigation of adaptive wavelets, based on spline techniques. A particular interest is in modeling by spline and radial basis functions and in constructing and analyzing the corresponding wavelets. In view of the fact that most modeling functions, including all the interesting standard radial basis functions, do not satisfy a two-scale (or refinement) relation, a unified scheme for the construction of wavelets that does not rely on such a relation is developed. This constitutes the first area of the proposed research. The second area is centered around multivariate wavelets, where the main concern is matrix dilation. For instance, the boundedness of the affine operator and that of the corresponding Littlewood-Paley energy functions, for a finite family of square-integrable functions with zero mean, will be investigated and sufficient conditions that guarantee lower bounds will be established. The third area is centered on the study of localized cosine Riesz bases and their corresponding duals. The duals, or bi-orthogonal bases, must facilitate computational efficiency. For this and other reasons, spline techniques, such as knot insertion and knot removal, will be investigated. The final area concerns the modeling of discrete data. For high dimensions, radial basis functions seem to be the most efficient tool, and the emphasis will be on various linear and nonlinear problems involving these functions. The central theme of this rese arch proposal is the development of mathematical tools for analysis and synthesis of discrete data in one or higher dimensions, in an Euclidean space or on a closed manifold such as the unit ball. This proposed research is a unified approach to represent any discrete data set in an effective way so that "wavelet" analysis can be readily performed. Both spline and radial basis functions will be used for data representation. It is well known that spline functions provide a very powerful tool for data modeling due to their most desirable properties such as flexibility for adaptive implementation, computational efficiency, and localization capability. On the other hand, radial basis functions are more powerful for handling higher-dimensional data sets, particularly when the data information is taken randomly. Wavelets will be constructed and wavelet algorithms will be developed by using both spline functions and radial basis functions. In addition, for analyzing nonstationary data that change in shape, the investigators will look into the use of localized cosine transforms by using splines or radial basis functions as envelopes (or windows). Adaptive algorithms will also be developed. One of the main reasons for wavelets to be considered as a very powerful tool for data analysis is their fast algorithms. The research will be focused on constructing wavelets using both spline and radial basis functions for effective implementation and fast computation. Software development will be carried out concurrently, and one of the main goals is to ensure that the algorithms can also be implemented in hardware. Industrial standards in the hardware prototype development will also be kept in mind.