基于压缩感知的信号重建快速算法研究

Compressed sensing has the extensively applied foreground in many fields such as video monitoring, SAR image, medical imaging. It is of great importance to establish rapid and effective algorithms of signal reconstruction. In the subject,fast methods for the L1-L2 nonconvex optimization model of sparse signal reconstruction are presented .First, we make the best of the sparse matrix vector multiplication and propose a fast Chaotic DCA algorithm based on non-convex sparse optimization model, combined DCA algorithm and generalized inverse techniques. And we discuss its convergence and steadiness. Then,the reweighted CDCA algorithm is proposed ,and new weight coefficient standard of rebuilding information is used to guarantee its fast convergence, and its convergence is proved. In the end, the equivalent relationship between the CDCA algorithm and the Bregman iteration for L1-L2 non-convex optimization model is proved , and the principle of fast convergence of these algorithms is given.

压缩感知在视频监控、遥感图像、医学成像等诸多领域中有着广泛的应用前景，建立快速有效的信号重构算法对其具有深远的意义。本课题针对信号稀疏重建的L1-L2非凸优化模型提出了几种快速算法。首先，将DCA算法与迭代广义逆技术结合，充分利用计算量为矩阵向量乘积的优势，提出了快速Chaotic DCA算法（CDCA），并讨论其收敛性及稳定性；其次，在CDCA算法基础上引入重复加权的思想，利用重建信息选取新的权系数指标，以保证算法的快速收敛，并给出算法收敛性的证明；最后，在理论上证明求解L1-L2非凸优化模型的CDCA算法与Bregman 迭代法之间的等价关系，给出算法快速收敛的理论原理。

本课题针对信号稀疏重建的非凸优化模型提出了几种快速算法。首先，结合求解L1的Chaotic算法，充分利用计算量为矩阵向量乘积的优势，提出了快速Chaotic 算法求解一般非凸优化模型Lp（p为参数），并讨论其收敛性及稳定性；其次，在此基础上引入重复加权的思想，利用重建信息自适应选取新的权系数指标（w-p），以保证算法的快速收敛，并给出算法收敛性的证明；最后，在理论上证明带有不同阈值算子的线性化Bregman 迭代法与权系数的选择之间的关系，给出相关快速收敛的理论原理。同时，尝试了将这些快速算法应用于组合非凸优化模型L1-L2，进行了初步的数值实验。