Multidimensional Multiwavelets and Time-Frequency Decomposition Techniques

摘要：本课题将研究两个主要主题。研究的一部分将解决小波理论中最困难的开放问题之一；即，发展了高维不可分正交小波和多小波构造的一般框架。本文的出发点是最近在任意膨胀矩阵下多维多尺度函数精度条件的时域PI表征。结合自相似性和迭代函数系统的见解，这表明了在一维结构中起关键作用的分解的弱替代的可能性。研究的第二部分涉及巴拿赫空间中的帧，特别是Gabor或窗口傅里叶帧。由代数和分析之间的相互作用引起的Gabor系统的问题将被解决，Gabor框架分解将被应用于分析伪微分算子和其他算子，利用Gabor框架为被称为调制空间的函数空间类提供无条件分解的原理。该项目涉及信号处理和分析的数学基础，如图像、音乐和语音，以及更抽象的信号类型。处理一个信号可以先把它分解成简单的基本单元，然后用几种不同的方法处理这些单元。这可以以非冗余的方式完成，其中每个基本单元独立于其他单元并包含整个信号的一个重要部分，或者可以以冗余的方式完成，其中包含在不同基本单元中的信息重叠。非冗余分解（“基”）可以提供信号的最紧凑的表示，而冗余分解（“帧”）允许重构信号，即使在传输中丢失了一些基本单元。本研究的目的是构建和应用这两种类型的分解在不同的设置。高维的小波基特别适合于图像分析。Gabor框架与数学、量子力学和信号处理中使用的分解有关，并将在本项目中应用于对信号的算子或变换的分析。

Proposal: DMS-9970524Principal Investigator: Christopher E. HeilAbstract: This project will pursue two main themes. One portion of the research will address one of the most difficult open problems in wavelet theory; namely, the development of a general framework for the construction of nonseparable orthogonal wavelets and multiwavelets in higher dimensions. The starting point is the recent time-domain characterization by the PI of the accuracy conditions for multidimensional multiscaling functions under an arbitrary dilation matrix. Combined with insights from self-similarity and iterated function systems, this suggests the possibility of weak substitutes for the factorizations that play the key role in one-dimensional constructions. The second portion of the research concerns frames in Banach spaces, especially Gabor or windowed Fourier frames. Issues for Gabor systems arising from the interaction between algebra and analysis will be addressed, and Gabor frame decompositions will be applied to analyze pseudodifferential and other operators, utilizing the principle that Gabor frames provide unconditional decompositions for the class of function spaces known as the modulation spaces.This project addresses the mathematics underlying the processing and analysis of signals, such as images, music, and speech, but also more abstract types of signals. A signal can be processed by first breaking it down into simple, basic units and then manipulating those units in several different ways. This can be done in a nonredundant fashion, where each basic unit is independent of the others and contains one essential part of the total signal, or it can be done in a redundant way, where the information contained in different basic units overlaps. Nonredundant decompositions ("bases") may provide the most compact representation of a signal, whereas redundant decompositions ("frames") allow reconstruction of the signal even if some of the basic units are lost in transmission. The objective of this research is to construct and apply both types of decompositions in various settings. Wavelet bases in higher dimensions are especially suitable for the analysis of images. Gabor frames are related to decompositions that have been used throughout mathematics, quantum mechanics, and signal processing, and will be applied in this project to the analysis of operators, or transformations, of signals.