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Representation Frames and Applications

表示框架和应用

基本信息

批准号：
1712602
负责人：
Deguang Han
金额：
$ 19.9万
依托单位：
The University of Central Florida Board of Trustees
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2021-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712602&HistoricalAwards=false
关键词：
Representation Frames Applications

项目摘要

In many engineering applications signals pass through linear systems, but in this process the recorded phase information can be lost or distorted. Examples of this problem occur in speech recognition, quantum state tomography, x-ray crystallography, and electron microscopy. The frame based phase retrieval problem is to recover a signal from the absolute values of its frame measurement coefficients. The central theme of this project lies in the investigation of the theory of phase-retrievable representation frames and its state-of-the-art applications in signal processing and information theory. The main objective is to establish the theoretical foundations for phase-retrievable representation frames and develop new representation frame based recovering algorithms that will address the computational cost problems in the existed recovering algorithms. The project will focus on three main problems in frame based phase retrieval. The first is representation frames and phase-retrieval. Due to their nice algebraic and/or geometric features, well-structured frames are excellent candidates for applications. This part of the project will focus on establishing the theory for finite group projective representation frames that admit phase-retrievable frame vectors. The goal is to completely characterize all such representations, and obtain simple and easily verifiable criterion for all the phase-retrievable frame generators so that they can be easily constructed for applications. The investigation on the representation phase-retrievable frames for ordered product of semi-groups is in line within the scope of representation frames and is directly targeting at applications such as dynamic sampling. The second project will focus on phase-retrieval for signals in the union of lower dimensional spaces. Many applications often face great computational challenges when recovering lower dimensional signals sitting in a very large dimensional Hilbert space. Building the theoretical foundation for the existence of phase-retrievable representation frames with much smaller length for such signals will greatly reduce the encoding frame length and obtain good characterizations for all such frames. The third project is on representation frames in erasure-corrupted signal recovering. Data transmission often causes erasures and other type distortions. In many cases the locations of erased frame coefficients are unknown, and/or the received partial data might be disordered. This project will investigate several practical problems with applications of representation frames in signal recovering from disordered partial frame coefficients and in the frame design problem for encoder-decoder protections.

在许多工程应用中，信号通过线性系统，但在这个过程中，记录的相位信息可能会丢失或失真。这个问题的例子出现在语音识别，量子态断层扫描，X射线晶体学和电子显微镜。基于帧的相位恢复问题是从信号的帧测量系数的绝对值恢复信号。该项目的中心主题在于相位可恢复表示框架理论的研究及其在信号处理和信息论中的最新应用。本文的主要目的是为相位可恢复的表示框架建立理论基础，并针对现有恢复算法中存在的计算量大的问题，提出新的基于表示框架的恢复算法。该项目将集中在基于帧的相位检索的三个主要问题。首先是表征框架和相位检索。由于其良好的代数和/或几何特征，结构良好的框架是应用的优秀候选者。该项目的这一部分将重点建立允许相位可检索框架向量的有限群投影表示框架的理论。我们的目标是完全表征所有这些表示，并获得简单，易于验证的标准，所有的相位可检索的帧生成器，使他们可以很容易地构造的应用程序。半群有序积的表示相位可恢复框架的研究属于表示框架的范畴，直接针对动态抽样等应用。第二个项目将集中在低维空间的联合信号的相位检索。当恢复位于非常大维度的希尔伯特空间中的低维信号时，许多应用经常面临巨大的计算挑战。为这类信号的相位可恢复表示帧的存在建立理论基础，将大大减少编码帧的长度，并获得所有此类帧的良好表征。第三个项目是关于擦除信号恢复中的表示帧。数据传输经常导致擦除和其他类型的失真。在许多情况下，擦除帧系数的位置是未知的，和/或接收到的部分数据可能是无序的。本计画将探讨几个实际问题，应用代表帧于从乱序部分帧系数中恢复信号，以及编码器-解码器保护的帧设计问题。