基于压缩感知的稀疏信号重建算法的理论研究

Compressed Sensing (CS) is an emerging area in signal processing domain these years. Compared with traditional sampling, CS integrates compression into the sampling process, which significantly reduces the data for transmission and storage. One of the important tasks in CS is how to recover the signals more accurately and effectively, which is concerned by many researchers... Focus on the practical problems that the special structure of the signals-block sparse signals, the major goal of this project is present some fast algorithms which are combined with the sparse regularization algorithm in learning theory. We discuss the theoretical foundation of these algorithms, the convergence criteria is also given. Simultaneously, we consider sparse recovery algorithm via overcomplete dictionaries , the inherent relations between compressed sensing and best k-term approximation is also discussed. This project belongs to the theory and application of reconstruction algorithms in compressed sensing. The study is of important theoretical value and practical significance.

压缩感知理论是近几年信号处理领域兴起的前沿课题。与传统采样不同，压缩采样把信号压缩融合到采样中，其采样频率可远低丁奈奎斯特频率，有效降低了信息传输、存贮的数据量。压缩感知的一个重要任务就是对压缩采样后的信号进行重构，目前引起了众多学者的关注和研究。. 本项目的主要目标是，结合学习理论中的稀疏正则化算法，针对实际问题中一类特殊结构的信号- - 块稀疏信号的特点, 提出若干种快速重构算法，研究算法的收敛性，并探讨算法的理论基础；同时开展基于冗余字典的稀疏信号恢复算法研究以及探讨压缩感知和最佳K项逼近的内在联系。本项目属于压缩感知中重构算法的理论及应用研究， 具有一定的理论价值和广泛的应用前景。

本项目的主要目标是，针对实际问题中具有一定结构信号的特点, 结合学习理论中的稀疏正则化方法，提出若干种快速重构算法，研究算法的收敛性，并探讨算法的理论基础；同时开展基于冗余字典的稀疏信号恢复算法研究以及探讨压缩感知和逼近理论的内在联系。项目的主要研究内容是，针对若干种典型问题，利用RIP条件对稀疏恢复算法进行收敛性分析，利用非凸优化的方法解决一类低秩矩阵以及稀疏信号的恢复问题，以及研究若干种典型恢复算法的GPU加速方法。本项目研究具有一定的理论价值和应用前景，其研究内容涉及信号处理、稀疏逼近等众多研究热点。