Highly Nonlinear Approximation: Theory and Algorithms

PI：Pencho Petrushev，南卡罗来纳大学DMS-0200665 ==摘要 该项目将侧重于大型冗余系统的非线性n项近似的具体问题，这通常被称为高度非线性近似。 冗余系统已经在信号和图像处理以及一般的近似理论中产生了严重的影响。研究界目前正在开发两种形式的冗余系统。第一个处理库的基础，如小波包和余弦包。第二个主要的推动力是研究字典，如Gabor函数和神经网络。 我们调查的重点是对字典的不同性质，其中分为两类：（一）字典组成的位移和膨胀的无限光滑，迅速衰减的功能，其中包括高斯，径向部分分数，或更一般的各向异性系统组成的某些变换等功能。 （二） 的多级并元划分生成的分段多项式的集合， 空间、多级嵌套三角剖分或单纯形分区。 该项目的主要目标是了解非线性和高度非线性过程。特别是，我们必须发展的光滑性条件（指定的特定空间）的性质的理解，它管理的速度在一个重要的度量集合的近似。该项目的第二个主要努力是开发基于基本分析的算法，能够实现最佳近似率（即压缩），并切实可行的实施。该项目正在积极协调其调查与合作者参与分析和介绍数据的广泛应用领域，包括由地理信息系统（GIS）、信号和图像分析、计算机辅助几何设计（CAGD）、图形绘制和大规模数值模拟的可视化产生的数字高程图和相关图像。在每种情况下，目标是在规定的字典的上下文中分析复杂的数据和信号，并将其分解为对所研究的物理系统来说是自然的基本元素。与基本元素相关联的幅度，无论它们是小波、多分辨率冗余系统还是各向异性系统，都提供关于数据的固有信息，并且理想地导致信号（数据）类的最大熵编码。数据的这种编码表示对于存储、传输、快速查询、视觉显示、相关性或与来自其他模态的数据的配准是重要的。本项目的主要动机是为这些调查的基本理论基础作出实质性贡献。

PI: Pencho Petrushev, University of South CarolinaDMS-0200665 ========ABSTRACT The project will be focused on specific problems in nonlinear n-term approximation from large redundant systems, which is commonly referred to as highly nonlinear approximation. Redundant systems have already had a serious impact in signal and image processing and upon Approximation Theory in general. The research community is currently developing two forms of redundant systems. The first deals with libraries of bases such as the wavelet packets and cosine packets. The second major thrust has been to study dictionaries such as the Gabor functions and neural networks. The focus of our investigations is toward dictionaries of a different nature, which fall in two categories:(i) Dictionaries consisting of shifts and dilates of infinitely smooth, rapidly decaying functions which include the Gaussian, radial partial fractions, or more general anisotropic systems consisting of certain transformations of such functions. (ii) Collections of piecewise polynomials generated by multilevel dyadic partitions of the pace, multilevel nested triangulations, or simplex partitions. The primary goal of project is to understand nonlinear and highly nonlinear processes. In particular, we must develop an understanding of the nature of the smoothness conditions (specified by specific spaces) which govern the rate of approximation in a collection of important metrics. The second major effort of the project is to develop algorithms which are grounded in fundamental analysis, capable of achieving the rate of the best approximation (i.e. compression), and are practical to implement.This project is actively coordinating its investigations with collaborators involved in the analysis and presentation of data for a wide range of application fields, including digital elevation maps and associated imagery arising from geographical information systems (GIS), signal and image analysis, computer aided geometric design (CAGD), graphical rendering and visualization of large scale numerical simulations. In each case, the objective is to analyze complicated data and signals in the context of prescribed dictionaries and to decompose them into basic elements which are natural for the physical system under investigation. The amplitudes associated with the base elements, whether they be wavelets, a multiresolution redundant system, or anisotropic systems, provide inherent information about the data and ideally lead to the maximal entropy encoding of classes of signals (data). This encoded representation of data is important for the storage, transmission, fast query, visual display, correlation, or registration against data from other modalities. The primary motivation of this project is to make substantial contributions to the underlying theoretical foundation for these investigations.