Representation Frames and Applications

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、量子状态断层摄影术、X射线结晶学和电子显微镜中。基于帧的相位恢复问题是从信号的帧测量系数的绝对值中恢复信号。这个项目的中心主题是研究相位可恢复表示帧的理论及其在信号处理和信息论中的最新应用。主要目的是为相位可恢复的表示帧建立理论基础，并开发新的基于表示帧的恢复算法，以解决现有恢复算法中的计算代价问题。该项目将重点研究基于帧的相位检索中的三个主要问题。第一个是表示帧和相位恢复。由于其良好的代数和/或几何特征，结构良好的框架是很好的应用候选者。项目的这一部分将集中于建立有限群射影表示框架的理论，该框架允许相位可检索的框架向量。目标是完全刻画所有这些表示，并获得所有相位可恢复帧生成器的简单且容易验证的准则，以便为应用程序容易地构造它们。关于半群有序积的表示相可恢复框架的研究是在表示框架的范围内进行的，并且直接针对动态抽样等应用。第二个项目将侧重于低维空间联合中的信号的相位恢复。当恢复位于非常大的维希尔伯特空间中的低维信号时，许多应用程序经常面临巨大的计算挑战。为这种信号存在长度小得多的相位可恢复表示帧建立理论基础，将大大减少编码帧长度，并获得对所有这样的帧的良好的表征。第三个项目是关于擦除损坏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，擦除的帧系数的位置是未知的，和/或接收的部分数据可能是无序的。本项目将研究表示帧在从无序的部分帧系数中恢复信号以及在编解码器保护的帧设计问题中的应用的几个实际问题。

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.