冗余字典下的压缩感知理论及应用研究

The flexibility of redundant dictionaries (or frames) is the key characteristic that empowers dictionaries to become a more natural and concise signal representation tool than bases. Therefore, compressed sensing that deals with sparse representations with respect to dictionaries becomes particularly important. This proposal will focus on the topic of compressed sensing with redundant dictionaries and try to solve the following three important problems. (1): We will develop the performance analysis of both standard L1-analysis and L1-synthesis under a unified framework. Then we exploit the relationship between the two approaches theoretically. Our goal is to solve an open problem in this field. (2): Based on the split Bregman iteration, we have proposed an effective iterative algorithm for solving the optimal-dual-based L1-analysis problem. We will go further to establish the convergence analysis of the proposed algorithm. Moreover, we try to develop some other (maybe more) effective algorithms to solving the optimal-dual-based L1-analysis problem. (3): We will apply the established theory and algorithms to solve some problems arising in image processing.

相对于基表示系统而言，冗余字典(或框架)的灵活性已使其成为一种自然而简洁的表示工具。因此，考虑冗余字典下的压缩感知将具有更加广泛的理论和现实意义，也是压缩感知领域重要的发展方向之一。本项目围绕冗余字典下的压缩感知展开研究，试图解决三个方面的问题。(1)：在统一的体系下建立经典L1-分析与L1-综合信号重构的误差界，从理论上分析它们之间的区别和联系，解决该领域的一个公开性问题。(2)：运用Split Bregman 迭代，我们已经提出了一种有效的算法求解基于最优对偶的L1-分析优化问题。我们拟进一步证明该算法的收敛性，分析其收敛速度，并试图寻找其他(或更为)有效的算法求解基于最优对偶的L1-分析优化问题。(3)：把建立的理论和算法应用到基于冗余字典的图像处理中去，解决实际应用中的具体问题。

相对于基表示系统而言，冗余字典(或框架)的灵活性已使其成为一种自然而简洁的表示工具。因此，考虑冗余字典下的压缩感知将具有更加广泛的理论和现实意义，也是压缩感知领域重要的发展方向之一。本项目围绕冗余字典下的压缩感知展开研究，解决了如下三个方面的问题。(1)：在统一的体系下建立L1-综合与经典L1-分析信号重构的误差界，并从理论上分析它们之间的区别和联系。(2)：基于Nesterov加速迭代的思想，提出了一种更为有效的迭代算法求解基于最优对偶的L1-分析优化问题，并把该方法应用于实际图像处理。(3)：从理论上建立了Shannon-Nyquist-Whittaker采样定理，不确定性原理和压缩感知之间的关系，并把该理论应用于量子信息中去。