结构化压缩感知及其在盲信号处理中的应用

Compressive Sensing (CS) is an new developed theoretical framework for information acquisition and processing, which is based on mathematics theory. CS is one of the hottest research topic in signal processing, system identification and pattern recognition. However, the conventional CS model consider the sparsity without the additional structure information. In fact, structured sparsity will lead better reconstruct result. This project is focus on structured compressive sensing problem, which is based on the research of block-sparse representation. The main contributions of this project are as follows: Firstly, based on the internal relationship between the sparse elements, we will build an unified and brief frame to measure the different sparse structure. Further more, we will build a new structured compressive sensing model by combining the proposed sparse measurement with compressive sensing model. Secondly, we will propose fast algorithms for large-scale structured compressive sensing. The proposed algorithms deal with the non-smoothness of the objective function in structured CS problem, which has low iteration complexity and suitable for large-scale problem since it just only use the first order derivative of the objective function. Finally, we will give a blind source denoising method in the presence of impulsive noise by the structrued sparse model. The proposed project has a certain potential theoretical and application value since it expand the theory and application range of structured compressive sensing.

压缩感知是建立在数学理论基础上的一种全新的信息获取与处理的理论框架，是近年来信号处理、系统辨识和模式识别领域的研究热点之一．传统的压缩感知模型中没有充分利用信号的稀疏结构这个先验信息．事实上，利用信号的稀疏结构可以得到更好的重构效果．本项目将在前期针对分块稀疏这一特殊的稀疏结构进行研究的基础上，对结构化压缩感知展开研究，主要研究内容包括：针对不同的稀疏结构，构造统一、简洁的稀疏度量函数，并将将新的稀疏度量函数与压缩感知框架结合，建立统一的结构化压缩感知模型；针对该模型设计快速算法，新算法将考虑结构化压缩感知问题中目标函数非光滑等特性，只利用目标函数的一阶导数信息，具有较低的计算复杂度，适用于大规模问题的求解；将结构化稀疏模型用在盲信号处理中，去除信号采集中出现的脉冲噪声，以更好的分离源信号和辨识信道．本项目的研究使得结构化压缩感知在理论和应用上都具有良好的扩展性，具有一定的理论和应用价值.

压缩感知是建立在数学理论基础上的一种全新的信息获取与处理的理论框架，是近年来信号处理、系统辨识和模式识别领域的研究热点之一．结构化压缩感知将稀疏结构与压缩感知框架结合，能够更好的描述信号的特征，得到更好的处理结果。 本项目围绕结构化压缩感知的模型，算法和应用进行研究。取得的研究成果如下：对目前已有的稀疏结构给出了一个统一的描述框架，并利用分割Bregman迭代算法求解该结构化稀疏重构问题；基于Moreau包络和最小最大凹罚函数，针对二维全变差去噪模型给出了其非凸正则，这些非凸正则同时可以保证目标函数整体的凸性，同时利用前向后向分裂算法求解新的目标函数；对目前常用的几种恢复算法进行理论分析，改进了其等距约束条件；将Contourlet变换作为压缩感知的稀疏变换基，并应用到图像重构中，新的变换基可以更好的重构图像的边缘和轮廓信息。..