Research On Algorithms Of Construction For Observation Matrix And Reconstruction In Compressed Sensing

The amount of data grows at explosive rate in the era of big data,but there is often a lot of redundancy in these data,removing redundancy in these data and compression with the requirement of restoration accuracy are the focus of research.In recent years,the theory of Compressed Sensing(CS)for sparse signal requires only a small amount of observation data to reconstruct the original signal with high precision,which reduces the sampling frequency,breaks through the limitation of the traditional Nyquist sampling theory,alleviates the limitation in hardware for sampling device and reduces the cost of data storage,processing and transmission.The construction of observation matrix and the algorithm for signal reconstruction are the important part of the research on the CS and are also studied in this article.The main work and the innovative points in this article are as follows:1.The basic architecture of CS theory is introduced briefly in this paper,then the constraint condition for the observation matrix is elaborated,so is the principle and the problem in the existing algorithms aimed to construct the observation matrix,the existing reconstruction algorithms are analyzed and then classified according to whether or not the prior information of sparsity is needed.2.In order to reduce the correlation between the observation matrix and the sparse basis so that the observation value can contain more information of the original signal and then the error for reconstruction can be reduced,the optimization of the Gram matrix and the construction of the observation matrix from the Gram matrix are studied.In the point of the Gram matrix optimization,the objective function of Gram matrix optimization is constructed according to the correlation of Gram matrix in CS can not always reach the bound of Welch;By obtaining the condition of the sensing matrix when the objective function takes the extreme value,combining the singular value decomposition of the sparse basis and the sensing matrix,the Gram matrix described by a single matrix variable is obtained;The above Gram matrix is substituted into the objective function,and the value of the single matrix variable is obtained by minimizing the objective function,then the analytic formula of the optimized Gram matrix is obtained.In the point of the construction for the observation matrix from the Gram matrix,based on the idea of updating the objective matrix row by row in the K-SVD algorithm,the square of the F norm for the difference between the target Gram matrix and the Gram matrix corresponding to the observation matrix is taken as the objective function,then a set of row vector in the observation matrix is obtained by solving the extremum of the objective function,the value of the row vector in the observation matrix is selected from the above set when the objective function takes the minimum based on the singular value decomposition of the error matrix,in other words the obtained row vector which minimizes the distance between the Gram matrix corresponding to the observation matrix and the target Gram matrix is obtained in the case where the remaining rows are fixed.3.An algorithm for the construction of observation matrix is proposed: in order to reduce the influence on the Gram matrix optimization form the initial value of the Gram matrix,based on the basis of the optimized Gram matrix analytic formula,the Gram matrix is iteratively optimized.The optimized Gram matrix processed by the threshold function is treated as the objective matrix in the objective function for the next iteration,and the iteration is terminated when the correlation of the optimized Gram matrix is less than the correlation of the target matrix.Based on the optimized Gram matrix mentioned above,the observation matrix is optimized iteratively using the obtained row vector analytic,whether the correlation of the Gram matrix corresponding to the optimized observation matrix is larger than the correlation of the target Gram matrix or not is used as the criteria for judging whether the iteration is terminated or not.The simulation results show the effectiveness of the proposed algorithm.4.In order to complete the signal reconstruction when the sparsity is unknown,a sparsity adaptive reconstruction algorithm is proposed: In the point of the selection from the columns of the observation matrix,based on the tree-shaped extended candidate sets in the MMP algorithm,the minimum of the residuals corresponding to all candidate sets in each iteration and the small threshold are compared to judge whether or not the iteration is terminated;Aiming at the problem that the number of candidate sets increases exponentially as the iteration goes on,which results in a large amount computation,the candidate sets are selected twice by the regularization and backtracking tracing.In order to balance the contradiction between the reconstruction precision and the computational complexity caused by the fixed expansion constant when the candidate sets are tree-shaped extended,a small expansion constant is used to reduce the computational complexity in the early stage of reconstruction,and a large expansion constant is used at the later stage of reconstruction to improve the reconstruction accuracy.The simulation results show the effectiveness of the proposed algorithm.