压缩感知中观测矩阵的优化算法

摘要

The observation matrix plays a key role in Compressive Sensing.Aiming to reduce the correlation between the observation matrix and the sparse base and then to improve the quality of the reconstruction,an optimal algorithm for observation matrix with the premise of a known sparse base was presented in this paper,the Frobenius norm of the difference between the Gram matrix and the matrix which was approximately close to Equiangular Tight Frame (ETF) was considered to be the objective function,then the optimization solution was conducted from the objective function to get the theoretical expression of the optimal Gram matrix,finally the iterative optimization was adopted to reduce the mutual coherence between the observation matrix and the sparse base,then the proposed optimal Gram matrix processed by the threshold function would be the ETF in the next iteration.It was shown in the simulation results that with the acceptable amount of computation,the reconstruction performance is better with the observation matrix produced in the proposed algorithm,especially when the signal's sparsity is high or the number of observations is few the reconstruction pedormance is much more better.%观测矩阵在压缩感知(CS)中起着关键性的作用.基于降低观测矩阵与稀疏基的相关性可以改善重构质量,本文提出了一种稀疏基已知的前提下观测矩阵的优化算法.该算法首先将Gram矩阵与逼近等角紧框架(ETF)的矩阵差的F范数作为目标函数,其次对目标函数进行最优化求解在理论上得到最优Gram矩阵的表达式,最后使用迭代优化来降低观测矩阵与稀疏基的相关性,将产生的最优Gram矩阵经阈值函数处理后作为下一轮迭代时目标函数中逼近ETF的矩阵.仿真实验表明,在可接受的运算量下,使用该算法优化后的观测矩阵可获得较好的重构效果,特别当信号稀疏度较高或者观测次数较少时重构效果的改善尤为明显.