Multidimensional Multiwavelets and Time-Frequency Decomposition Techniques

建议：DMS-9970524首席调查员：Christopher E.海尔摘要：本项目将探讨两个主要主题。这项研究的一部分将解决小波理论中最困难的公开问题之一，即发展构造不可分的正交小波和高维多小波的一般框架。本文的出发点是多维多尺度函数在任意伸缩矩阵下用PI对精度条件的最新的时间域刻画。结合来自自相似和迭代函数系统的见解，这表明在一维结构中起关键作用的因式分解有弱替代的可能性。第二部分研究了Banach空间中的框架，特别是Gabor或加窗傅立叶框架。本课程将讨论由于代数和分析之间的相互作用而产生的Gabor系统问题，并将利用Gabor框架为称为调制空间的一类函数空间提供无条件分解的原理，将Gabor框架分解应用于分析伪微分和其他算子。该项目解决了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。可以通过首先将信号分解成简单的基本单元，然后以几种不同的方式处理这些单元来处理信号。这可以以非冗余的方式来完成，其中每个基本单元独立于其他单元并且包含总信号的一个基本部分，或者可以以冗余的方式来完成，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表示，而冗余分解(帧)允许重构信号，即使一些基本单元在传输中丢失。本研究的目标是在不同的环境下构建和应用这两种类型的分解。高维小波基特别适用于图像分析。Gabor框架与已在整个数学、量子力学和信号处理中使用的分解相关，并将在本项目中应用于对运算符或信号变换的分析。

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.