基于优化观测矩阵的共轭梯度改进算法

Improved Conjugate Gradient Algorithm Based on Optimization of Measurement Matrix

测量矩阵的设计和信号的重构是压缩感知理论研究的核心问题。基于梯度下降法的QR分解观测矩阵优化使得信号在观测过程中的主要信息得以保存,而共轭梯度法则在信号的重构性能方面较理想。为此,将观测矩阵优化引入到共轭梯度重构算法中,针对共轭梯度重构算法,基于梯度下降法的QR分解优化观测矩阵,得到一个新的重构算法,即基于优化观测矩阵的共轭梯度算法。改进的算法中矩阵具有较大的列独立性以及与稀疏矩阵间相关性较低的特性,同时具有较好的性能。利用Matlab仿真实验来验证共轭梯度法与观测矩阵优化结合的重构算法的可行性及有效性。仿真结果表明,基于优化观测矩阵的共轭梯度算法在运行时间上缩短了2~3倍,优于其他一些重构算法。

The design of measurement matrix and the reconstruction of signal is the key in the study of compressed sensing theory. QR de- composition measurement matrix optimization based on gradient descent method makes it possible to preserve the main information during the process of observation, and the conjugate gradient algorithm is ideal for the reconstruction of the signal. The measurement matrix has been introduced to optimize the conjugate gradient reconstruction algorithm. In view of the conjugate gradient reconstruction algorithm, QR decomposition based on gradient descent method is used to optimize the measurement matrix and a new reconstruction algorithm, a conjugate gradient algorithm based on optimization of the measurement matrix, is obtained. In the improved algorithm, the matrix has larger column independence and lower correlation with sparse matrix, and has better performance with conjugate gradient method. Simulation experiments with Matlab have verified the new algorithm is feasible and effective. The results show that the conjugate gradient optimiza- tion algorithm with measurement matrix has been reduced 2 - 3 times in running time, which is greatly superior to other reconstruction algorithm.