稀疏信息处理中的采样矩阵重置问题研究

Compressive sensing is a new proposed theory for signal acquisition and processing, it breaks through the traditional Shannon-Nyquist sampling law. It has been successfully applied in many fields such as satellite communications, remote sensor, pattern recognition, and image compression. Compressive sensing is closely related to the mathematical subjects such as matrix computation, operation and optimization, probability theory, functional analysis and so on. The design of the measurement matrix is a critical problem in compressive sensing since it directly relates to the quality of the recovery signal. Therefore, it has been extensively studied. However, the measurement matrix is determined by the hardware. It is difficult to find the hardware corresponding to the measurement matrices which have already been designed. For the above reason, we propose the problem of resetting the measurement matrix. By the optimized method, the demand of the hardware can be reduced and the applicability problem may be overcome. This project will intensively study the proposed problem, establish a series of theories and methods for resetting the measurement matrix, and provide the important theoretical foundation for the practical application.

压缩感知是近年来提出的一种新的信号获取与处理理论，它突破了传统的Shannon-Nyquist采样定律。在卫星通信、遥感成像、模式识别和图像压缩等诸多领域，已得到了成功应用。压缩感知与数学中的矩阵计算、运筹学与优化、概率论以及泛函分析等学科密切相关。采样矩阵的设计是压缩感知中的一个关键问题，它直接关系着重构信号的质量，因此得到了广泛关注。但采样矩阵由硬件设备决定，对于目前已设计出的大部分采样矩阵，很难有相应的硬件设备满足条件。鉴于此，本项目提出了采样矩阵的重设置问题，使用优化方法，降低了对硬件设备的需求，使上述可应用性问题的解决得到了可能。本项目拟对所提出的问题进行深入研究，将建立采样矩阵重设置问题的一系列理论和方法，为实际工程应用提供重要的理论基础和依据。

压缩感知是近年来提出的一种新的信号获取与处理理论，它突破了传统的Shannon—Nyquist采样定律。在卫星通信、遥感成像、模式识别和图像压缩等诸多领域，已得到成功应用。本项目对压缩感知的数学理论和应用进行了深入研究。主要结果有：给出了矩阵Spark的两个下界估计，并证明了这两个下界比目前已有的下界更精确；使用交替投影法，给出了稀疏信号恢复的一个新方法，并建立了该方法收敛的两个充分条件；首次将压缩感知理论应用于损伤识别问题中，其所需采样点的个数远少于目前已有的方法。