Representation Frames and Applications

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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.