压缩感知中正交匹配追踪算法的理论研究

Compressed sensing promotes the development of the theory and engineering application and has been one of the hottest topics in the field of signal processing. The reconstruction algorithm is an important part of compressed sensing. The project analyzes orthogonal matching pursuit (OMP) algorithm that has low complexity, the contents are as follows: ① under the case of noiseless, we improve the upper bound of the sufficient condition, which guarantees that OMP algorithm accurately reconstructs sparse signals, and get relaxed bound. Thus, reducing the number of sampling and the cost. ② under the perturbations of additive noise and multiplicative noise, this project researches the performance of OMP algorithm and obtains sufficient condition that has nothing to do with the unknown sparse signal. The condition will guide the engineering practice. In order to suppress the corruptions and noises, to ensure the recovery reliably and to reduce the number of the observed measurements at the same time, in the research, some mathematical methods and machine learning methods will be used, such as linear and nonlinear functional analysis, the space correlation analysis, numerical analysis, stochastic analysis, probability and statistics and sub-supervised or unsupervised learning methods. The proposal intends some influential achievements both in the theory and technology with high performance.

压缩感知促进了信号处理理论以及工程应用的发展，已经成为信号处理领域研究热点之一。重建算法是压缩感知理论的重要组成部分。本项目针对复杂度较低的正交匹配追踪算法（Orthogonal matching pursuit, OMP）展开研究，其内容如下：① 无噪的情形下，改进保证OMP算法精确重建稀疏信号的充分条件的上界，得到宽松的上界，进而减少采样数目，降低成本；② 加性噪声与乘性噪声干扰的情况下，针对OMP算法的重建性能展开分析，得到与未知稀疏信号无关的充分条件，进而对工程实践起到一定的指导作用。为了同时达到抗干扰，准确重建原信号及降低采样数目的目标，在研究中将采用线性与非线性泛函分析、空间相关理论、数值分析、随机分析、概率统计等数学理论方法和机器学习方法，以期在基本理论以及高性能关键技术方面取得较好的成果。

压缩感知（compressed sensing，CS）能够将高维的信号从低维的测量中精确重构，它充分利用了信号的内部结构，使得运用远少于香农采样定理中所界定的采样数目就可以精确重建原始信号。它具有深远的应用前景及意义。一经提出立即受到广大专家学者的广泛关注，并将其应用到诸多领域中。例如：压缩成像；超宽带信号处理；压缩机器学习；生物传感；雷达遥感；地球物理数据分析等等。.该项目主要研究了：①正交匹配追踪(orthogonal matching pursuit, OMP)的重建性能。在无噪的情况下，给出了能够保证该算法精确重建信号的充分条件。还给出了重建所需的必要条件。在加性噪声以及乘性噪声的干扰下，给出了保证精确重建信号支撑集的充分条件。通过举出反例，给出了重建所需的必要条件。这些条件对原有文献中的条件都有所改进。从而能够使得进一步降低采样率成为可能。②子空间追踪(subspace pursuit, SP)及压缩采样匹配追踪(compressive sampling matching pursuit, CoSaMP)的重建性能。在无噪及有噪的情况下，分别给出了保证这两种算法精确重建信号的充分条件。在感知矩阵为随机矩阵时，给出了保证重建信号所需的采样数目，该采样数目与原有文献中给出的采样数目有显著的降低。并且，在感知矩阵为随机矩阵及稀疏矩阵时，给出了重建随机信号的数值实验，说明这两种算法在噪声下的重建性能依然很好。这为以后的工程应用指明了方向。③加权L_{2,1}最小化方法的重建性能。在图像序列中，信号的支撑信息可以预先知晓，这就为进一步降低采样数目提供可能。利用这类已知的支撑信息，提出了加权L_{2,1}最小化方法解决larynx image sequence的重建问题。给出了保证该方法精确恢复信号的充分条件。与以往文献的方法不同，我们将图像看做矩阵来处理，大大的缩短了重建时间。