中文摘要：

压缩感知包括压缩采样与稀疏重构．压缩采样突破了传统的香农采样定理限制，降低了采集数据量，是新兴的信号采集方法．稀疏重构算法是恢复原始高维信号的关键步骤，已成为信号处理及相关领域的研究热点．设计了一种稀疏重构算法FPSP3，该算法包含3个要素：不动点迭代，SPG2非单调线搜索及热启动技术．将非光滑L1范数罚最小二乘的最优解表示为梯度算子与次微分算子和的零点，采用前向后向算子分裂法推导出最优解方程为包括前向梯度步与后向邻近步的不动点迭代，通过证明后向邻近步对应L1范数的邻近算子即软阈值收缩，从而将不动点迭代表示为梯度下降与软阈值收缩．通过证明梯度算子逆是强单调的从而简化了收敛步长分析，给出了不动点迭代线性收敛于最优解的简要证明．采用SPG2非单调线搜索与热启动技术显著加快了算法实际运行速率，在稀疏重构实验中与某些著名的L1范数方法进行了比较，结果表明FPSP3具有运算速度与重构精度优势．

英文摘要：

Compressive sensing consists of compressive sampling and sparse reconstruction. Compressive sampling breaks through the limitation of traditional Shannon sampling theorem, reduces the acquisition data amount hence becoming an emerging method for data acquisition. Sparse reconstruction algorithm is the key step for successfully recovering the original data in high dimension and has become the hot research topic in signal processing and related area. In this paper, an sparse recovery algorithm is designed and named FPSP3, which includes three key components： fixed point iteration, non-monotone line search SPG2 and warm start skill. The optimal solution of LI norm penalized Least Square problem is represented as the zero point of sum of gradient and sub-differential operator, and forward backward operator splitting method is used to derive the optimal solution＇ s fixed pointiteration, which consists of forward gradient and backward proximal step. The proof of backward proximal step corresponding to the proximity operator of L1 norm, namely soft thresholding shrinkage, demonstrates that fixed point iteration can be implemented by gradient descent and soft thresholding. The analysis for convergent step size condition is simplified by proving the strong monotonicity of inverse of gradient operator. The brief proof is given to illustrate the fixed point iteration converges in linear rate to the optimal solution. The introduction of non-monotone line search SPG2 and warm start skill significantly accelerate the practical efficiency of proposed algorithm. Comparisons are made in sparse recovery experiments with some state of the art L1 norm methods, which demonstrates the advantages of running time and recovering accuracy of FPSP3.