Research On Algorithm For L₁ Regularized Least Squares Problem In Compressed Sensing

Compressed sensing breaks through the original sampling theorem in a way.It can be subjected to fewer measurements than traditional methods to sample sparse signal,and compress the sampled data at the same time.Now,compressed sensing has been widely used in signal reconstruction,image processing,wireless sensor networks,medical imaging,nuclear magnetic resonance imaging and other fields.The signal reconstruction algorithm is one of the key contents of the compressive sensing theorem,and the algorithm for the l₁ regularized least squares problem plays an important role in the signal reconstruction algorithm.In this paper,we study some partial algorithms for solving the l₁ regularized least squares problem.Based on this,we improve the stepsize in the BB algorithm proposed by Huang（Huang_BB）and iterative approximation gradient projection（IAGP）algorithm,and respectively,propose a new sparse optimization（NSP）algorithm and a modified iteratively approximated gradient projection（MIAGP）algorithm.Specifically,the main contents of this paper are as follows:Firstly,this paper introduces the theoretical framework of Compressed sensing,and analyzes in detail these main aspects:the sparse decomposition,the observation matrix and the signal reconstruction.Then on the base of these,this paper studies some classical algorithms which can solve l₁ regularized least squares problem.Secondly,by researching IAGP algorithm,we improve the formula of stepsize to reduce the reconstruction time of IAGP algorithm in this paper.By designing a new approximate matrix of Hessian matrix,we obtain a new quadratic approximate model of function in the current iteration point,and use this model and retard technique to derive a new stepsize.Based on the new stepsize,we propose MIAGP algorithm for solving the problem of sparse signal reconstruction in compressed sensing.Convergence analysis of the algorithm is given,and simulation experiments are carried on.The experimental results show that the improved algorithm improves the reconstruction speed,and retain the advantage of IAGP algorithm.Finally,by researching Huang_BB algorithm,same as the stepsize method in MIAGP,use the quadratic approximate model of function in the current iteration point and retard technique to derive another new stepsize.Based on the new stepsize,we propose NSP algorithm.Convergence analysis of the algorithm is given,and simulation experiments are carried on.Results demonstrate that the improved algorithm can effectively reduce the number of iteration steps and improve the reconstruction speed.