Research On Sparse Representation Based Signal Recovery And Dictionary Learning Algorithms

In many branch fields of signal processing,measurement models can be expressed as lin-ear systems,solving them is precisely the signal recovery.However,system matrices modeled according to practical applications are generally rank-deficient or ill-conditioned,which makes solving such linear systems been ill-posed or ill-conditioned problems,thus solutions are non-unique or instable.To obtain meaningful solutions,introducing the prior information on original signals is required.In signal processing,sparse representation is a prevalent prior model,and the dictionary plays a significant role in it.This dissertation conducts the research on sparse representation based signal recovery and dictionary learning algorithms,not only providing im-proved versions or new convergence analyses for some existing algorithms,but also proposing new algorithms.These algorithms are mainly based on the synthesis model,including its three forms,i.e.,traditional sparsity,convolutional sparsity and double sparsity,while the analysis model is also involved.The contributions are as follows:·Since the convergence condition of the over-relaxed monotone fast iterative shrinkage-thresholding algorithm(MFISTA)is complex and not convenient to check,an improved version is proposed.The theoretical analysis demonstrates that a simple sufficient condition guarantees the convergence of the improved version.Moreover,experiments show that the improved ver-sion can accelerate MFISTA under some cases where system matrices are rank-deficient and ill-conditioned,while the original version cannot.·Leveraging sequential subspace optimization(SESOP)to accelerate MFISTA is proposed,and an adaptive method that sets the number of previous propagation directions in the subspace is proposed.Experiments show that leveraging SESOP has very obvious acceleration effect-s for MFISTA.The accelerated algorithms are based on MFISTA,thus the convergences are guaranteed.·Currently,the non-smooth version of iterative sparsification projection(ISP)algorith-m framework lacks proofs of the convergence.Therefore,for one of its algorithm instances,termed ISP-soft,a convergence analysis based on the fixed point mapping is provided.Numer-ical experiments verify some of the given convergence results.·Although the convergence of the orthogonal matching pursuit algorithm has been veri-fied,its convergence study is still insufficient under the case of the convolutional sparse coding.Therefore,two more refined new tools,the stripe coherence and the stripe restricted isometry property,are leveraged to give improved convergence analyses,deriving profounder conver-gence conditions and a tighter error upper-bound.·The double sparsity model based sparse dictionary learning is modeled as an uncon-strained bi-convex optimization problem,an algorithm based on the block proximal gradient framework is derived to solve this problem,the convergence analysis and the complexity anal-ysis of the algorithm are also provided.Experiments show that the proposed algorithm outper-forms two sparse dictionary learning algorithms given by previous works.·After some researches on the synthesis model,the analysis model is beginning to be paid attention.An algorithm is proposed for effectively solving the relaxed analysis least absolute shrinkage and selection operator(RALASSO)optimization problem.Further,not limited to the sparsity regime,thel₁-norm regularizer is replaced by a general regularization function,an algorithm framework is derived to solve this general optimization problem.Experiments show the superior performance of the proposed algorithms.