Abstract: In order to solve the problem of large correlation coefficients when obtaining the observation matrix from the optimized Gram matrix in Compressive Sensing (CS), based on the optimized Gram matrix obtained in the existing algorithm, the value of the row vector in the observation matrix when the objective function takes the extreme value was obtained based on the extreme value of the equivalent transformation of the objective function, the analytic formula of the row vector when the objective function takes the extreme value was elected from the values mentioned above by Singular Value Decomposition (SVD) of the error matrix, then a new observation matrix optimization algorithm was put forward by using the idea of optimizing the target matrix row by row in the K-SVD algorithm, the observation matrix was optimized iteratively row by row, and the difference between the correlations of the observation matrix generated by adjacent two iterations was taken as a measure of whether or not the iteration is completed. Simulation results show that the relevance between the observation matrix and the sparse base in the improved algorithm is better than that in the original algorithm, thus reducing the reconstruction error.

Key words: Compressive Sensing (CS), observation matrix, Singular Value Decomposition (SVD), analytic formula of row vector, iterative optimization

摘要： 针对压缩感知（CS）中从优化后的Gram矩阵求解观测矩阵时会出现较大相关系数的问题，在利用现有算法得到优化后的Gram矩阵的基础上，通过求解等价变换后的目标函数对观测矩阵行向量的导数得到目标函数取极值时行向量的值，并通过对误差矩阵进行奇异值分解（SVD）在上述行向量的值中选出使得目标函数取最值时行向量的解析式，在此基础上给出了观测矩阵的优化算法：通过借鉴K-SVD算法中逐行优化目标矩阵的思想，对观测矩阵进行逐行迭代优化，并将相邻两轮迭代产生的观测矩阵所对应的相关性之差作为衡量迭代是否结束的条件。仿真结果表明：该算法在观测矩阵与稀疏基的相关性方面优于改进前的算法，从而提高了重构精度。

关键词: 压缩感知, 观测矩阵, 奇异值分解, 行向量解析式, 迭代优化