Algorithm And Analysis Of Signal Recovery Based On Sparse Structure

Compressed sensing is a new type of sampling theory.The signal can be success-fully reconstructed using the sparseness or compressibility of signals,and the number of samples required is much less than traditional methods.The difficulty lies in con-structing effective recovery algorithms.A new recovery algorithm is proposed in this paper,and the effectiveness and priority of the algorithm are verified by numerical experiments.Moreover,The recovery performance of the orthogonal least squares al-gorithm in the noise case and the general perturbations case is analyzed.We study the sufficient conditions of unconstrained l1-l2 minimization to recover signals,and also consider the quadratically constrained basis pursuit(QCBP),Dantzig selector and Lasso estimator.The main research content of the dissertation are as follows.Firstly,based on the classical orthogonal least squares(OLS)algorithm,a new greedy algorithm-multipath least squares(MLS)algorithm-is proposed.The algorith-m uses tree-search structures to search multiple promising candidates in each iteration,increasing the chances of selecting the correct candidate.Based on the restricted isom-etry property(RIP),sufficient conditions for the MLS algorithm to accurately recover sparse signals are derived.In addition,a counter-example is constructed to show that this recovery condition is almost optimal.It is further considered that the MLS algo-rithm can still accurately recover the support of sparse signals in the presence of noise.Simulation experiments show that compared with other recovery algorithms,the MLS algorithm can effectively improve the signal recovery performance.However,the MLS algorithm is not yet perfect and has the problem of high computational complexity.In view of this problem,this paper proposes a modified form of MLS algorithm——depth-first MLS algorithm(MLS-DF).After experimental verification,compared with the MLS algorithm,the execution time of the algorithm is significantly shortened,al-though it will be slightly lost a little recovery performance,but still better than other algorithms.Secondly,we consider the support recovery of sparse signals in the case of noise with OLS algorithm.Based on the restricted isometric property,the OLS algorithm accurately recovers the support of sparse signals in the case of noise.The general perturbations case of OLS is further discussed,i.e.,the sensing matrix and observations are contaminated at the same time,and almost sparse signals can accurately recover the support of the best K-term approximation of the signal under some conditions.Thirdly,we consider the signal recovery conditions of unconstrained l1-l2 mini-mization method for DS-type noise and l2 bounded noise,respectively.This is the first time that unconstrained l1-l2 minimization is analyzed based on mutual coherence,supplementing the theory of unconstrained l1-l2 minimization for recovery signal.Finally,the conditions of three models of quadratically constrained basis pursuit(QCBP),Dantzig selector(DS)and Lasso estimator can be used to stably recover the signal in the presence of noise.Recovery conditions based on the cumulative coherence for QCBP and DS are the same as those obtained by l1 minimization for the noiseless case,which is also the best recovery condition at present.Moreover,a recovery guarantee for Lasso greatly improves the existing results.