Research On Low-Rank And Sparse Recovery Theory And Its Applications In Signal Processing

The theory of Low Rank and Sparse Recovery(LRSR)is to separate the low-rank part and the sparse part from an observation matrix with missing or corrupted data by exploiting the low-rankness or sparsity of the matrix.This method is based on structural clustering and optimization of sparse representations by considering the data matrix as the sum of a low-rank matrix and a sparse noise matrix.And recovering the low-rank matrices by solving the nuclear norm optimization problem.In this paper,onedimensional speech signal processing and two-dimensional image processing are used as application backgrounds,with low-rank and sparse recovery as the theoretical basis.Different low-rank and sparse algorithms are proposed based on the characteristics of signals in different application scenarios,and the effectiveness of the algorithms is verified through experiments.The main innovations of this paper are as follows:(1)In response to the problem of significant estimation error in traditional Direction of Arrival(DOA)estimation algorithms under low signal-to-noise ratio and limited snapshot conditions,this paper proposes an adaptive weighted low-rank and sparse recovery-based DOA estimation method.This method is based on LRSR theory and introduces an adaptive weighted nuclear norm.The model about the signal covariance matrix and noise covariance matrix is constructed and solved using the Alternating Direction Method of Multipliers(ADMM)method.To obtain more accurate solutions,this paper also proposes an adaptive termination condition method based on the normal distribution property of the covariance error.Finally,based on the reconstructed low-rank and noise-free covariance matrix,the DOA estimation is obtained using Multiple Signal Classification(MUSIC)algorithm.(2)In the theory of LRSR,the use of nuclear norm as a rank function in convex relaxation methods results in performance loss,which further leads to a decline in DOA estimation performance.To address this issue,this paper proposes a DOA estimation method based on adaptive weighted truncated nuclear norm.The algorithm combines non-convex γ norm,adaptive weighted nuclear norm,and truncated nuclear norm models to more accurately reconstruct the received signal covariance matrix as the sum of a low-rank signal covariance matrix and a sparse noise covariance matrix.In addition,the Generalized Alternating Direction Method of Multipliers(GADMM)with an adaptive stopping criterion is used to solve this convex optimization problem.Finally,based on the reconstructed low-rank noise-free covariance matrix,the DOA estimation is obtained using MUSIC.(3)A structure-aware adaptive reweighted low-rank and sparse recovery image denoising algorithm is proposed to address the poor performance of traditional low-rank and sparse recovery theory in high-density sparse noise image denoising.This algorithm introduces Total Variation(TV)regularization constraint into the adaptive reweighted low-rank and sparse recovery model,which not only improves the denoising effect but also effectively reduces the loss of image information while preserving the structural information in the original image.This convex optimization problem is solved by using ADMM and Fast Gradient Projection method(FGP).Experimental results show that the proposed algorithm has significant advantages in dealing with high-density sparse noise images.