Research On Matrix Optimization And Fast Reconstruction Algorithm In Compressed Sensing

Traditional sampling theorem requires sampling frequency should be higher than the double of signal bandwidth,so as to ensure the recovered signal will not be distorted.When it is in dealing with the broadband signals,the sharp increase of sampling frequency will bring massive data processing and transmission problems,which caused great pressure to the current information technology.The theory of Compressed Sensing(CS)provides a possible way to overcome this problem.It points out that if the signal is sparsely,it can be projected by the measurement matrix to get a small number of observation points,which can reduce the dimension of the signal from N dimension to M dimension(M is much smaller than N).This means that the use of CS technology is not limited by the signal bandwidth under certain conditions,but more related to the characteristics of signal itself.And the original signal can be recovered from these observation points by the correlation reconstruction algorithm.In this paper,the main work is to study the measurement matrix optimization and the sparse reconstruction algorithm in CS theory,the concrete content is as follows:(1)Describe the basic principle of compressed sensing technology,and briefly introduce the commonly used measurement matrix,signal reconstruction model and corresponding reconstruction algorithm.The construction conditions of measurement matrix are deeply studied.The measurement matrix is analyzed from the constraints of sampling independence,RIP characteristics and white noise after compression and it is proved that the measurement matrix has the best characteristics under the condition of equiangular tight frame(ETF).Finally,several methods to optimize the measurement matrix are introduced.(2)A new method to optimize the performance of measurement matrix is proposed: SO-QR method,which reduces the coherence coefficient of the recovery matrix by iteratively optimizing the column vectors with high correlation in the recovery matrix by schmidt orthogonalization.The approximate QR decomposition method is also used to optimize the measurement matrix in the iterative process.Simulation results show that the SO-QR method can reduce the correlation between the measurement matrix and sparse dictionary,improve the performance of the measurement matrix.(3)The sparse reconstruction algorithms in compressed sensing are studied in detail.Some typical greedy algorithms such as OMP,ROMP,CoSaMP,SAMP are briefly introduced,including their algorithm principle and implementation process,and their advantages and disadvantages.Finally,we propose an improved sparse adaptive compressive sampling matching pursuit algorithm.It can be used to reconstruct the sparse unknown signal accurately,and its performance is better than those algorithms described above.(4)Present a deterministic measurement matrix which is simple and easy to construct.Based on this,we propose a fast reconstruction algorithm that performs a thresholding in the DCT domain.The algorithm abandons the high-frequency components in the DCT domain which are insignificant to the signal,and selects the first M columns of the recovery matrix in the reconstruction process,and then directly solves the equations to obtain the sparse vectors,which can quickly reconstruct the original signal.Simulation results show that the proposed fast algorithm is much faster than OMP,StOMP algorithm,and the reconstruction result is better.