Research On Hard Thresholding Algorithms In Compressed Sensing Problems

Compressed sensing has attracted much attention because of its important applications in signal recovery.Hard Threshold Pursuit(HTP)can reconstruct sparse signal and plays an important role in solving compressed sensing problems.In this paper,two new iterative algorithms of hard threshold class are designed based on HTP,and theoretical analysis and simulation results are given to verify the effectiveness.The classical HTP uses only once hard thresholding operator to generate the subspace,and then solves the minimum value of the objective function in the subspace.Based on the observation that the hard thresholding operator should be applied to a compressible vector,we propose the Compressive Hard Thresholding Pursuit(CHTP)by introducing a compressive step to HTP,in which the new iteration point is obtained through twice hard thresholding operator and pursuit.According to the restricted isometry property of measurement matrix,a sufficient condition is given to ensure the convergence of CHTP.The iterative number and stability of CHTP are analyzed.The numerical experiments are carried out to compare classical algorithms,including HTP,Orthogonal Matching Pursuit(OMP),Subspace Pursuit(SP)and Compressed Sampling Matching Pursuit(Co Sa MP).The results show that the reconstruction capability and running time of CHTP are comparable to those of the above mainstream algorithms.The classic HTP directly generates the next iteration point from the current iteration point through the negative gradient direction.We can use the information of the current point and the previous point together to improve the performance of HTP.Thus we present a novel idea of using a convex combination of the current point and the previous point to generate an intermediate point.The intermediate point is used to generate subspace through the hard thresholding operator.At the same time,the pursuit step is added.We call Modified Hard Thresholding Pursuit(MHTP)for the new iteration format.Theoretically the convergence,the number of iterations and the stability of MHTP are analyzed.In numerical experiments,we compared MHTP with HTP,SP,Co Sa MP and Generalized Orthogonal Matching Pursuit(g OMP).The results show that MHTP has certain advantages in recovery success rate and running time.In addition,some examples of real image reconstruction is added,which fully demonstrates the effectiveness of the algorithm.In this paper,CHTP and MHTP which we proposed is modified and improved for HTP,which enriches the hard thresholding algorithms and provides more ideas for solving compressed sensing problems.