Research On Measurement Matrix Optimization And Signal Reconstruction In Compressed Sensing

Using the sparsity of signals,Compressed Sensing(CS)theory samples signals at a sampling frequency which is much lower than Nyquist sampling frequency through measurement matrix.With a small amount of measurement data,the original signal can be recovered precisely by reconstruction algorithm.The CS theory breaks through the restriction of Nyquist sampling theorem,and provides a solution to alleviating the storage pressure of data transmission and high speed sampling,which are common in traditional signal processing technology.Measurement matrix optimization and signal reconstruction algorithm are two important parts of CS theory.This article studies the above aspects separately,and the main work and innovations are as follows:(1)The basic principles and mathematical model of CS are introduced,and the constraints of the measurement matrix are described.The existing research results of the measurement matrix optimization algorithm and greedy reconstruction algorithm are introduced.The problems and shortcomings are emphatically ananlyzed,which exist in the optimal algorithm of measuement matrix based on gradient descent method and OCMP algorithm.(2)Aiming at the problems of slow convergence speed and low optimization accuracy caused by the gradient descent in Abolghasemi algorithm,a measurement matrix optimization algorithm based on BFGS method is proposed.First,the non-diagonal elements of Gram matrix are constrained by threshold function to obtain the objective matrix,and an objective function is contructed.Second,to solve the objection function,the approximate Hessian matrix is obtained to determine the search direction through BFGS method,and the Armijo criterion is used to determine the search step length.The effectiveness of the proposed algorithm is verified by simulation experiments.(3)Aiming at the problems of high time and low reconstruction accuracy of OCMP algorithm which are caused by the atomic selection mechanism in iteration and the single incremental expansion way of supporting sets,an improved OCMP algorithm for atomic selection strategy is proposed.To reduce the iteration time of the extended support set,the method of selectiong multiple matching atoms through the fuzzy threshold is proposed in each iteration.To improve the accuracy of supporting set atoms,the backtracing method is used.To make the improved OCMP algorithm adpted to the case of unknown sparsity,matching test method is used to estimate signal sparsity,then the improved OCMP algorithm is used to reconstruct the signal.The sparsity estimation is corrected by the accuracy of reconstructed signal in every iteration.The simulation results show that the above algorithm can reconstruct signal well in the corresponding case.