| The last decade has witnessed a growing interest in the research on Compressed Sensing (CS), a novel sampling paradigm introduced by Donoho, Candes, et. al.. Although CS was proposed in 2006, today it is still an attractive topic in the signal processing community and has found many emerging applications. As an important component of CS, many classes of interesting recovery algorithms have been developed, but, it is still an important issue to explore effective and robust recovery schemes.This dissertation aims at providing an investigation to solve the optimization problems with non-convex sparsity inducing penalties, i.e. so-called non-convex compressed sensing. It has been proved that non-convex compressed sensing has many potential advantages over the conventional convex relaxation algorithms, i.e. Basis Pursuit. However, such an algorithm cannot guarantee to converging to the global optimum in recovery procedure, during computation procedure, it may fall into the wrong basin of attraction. It is one of the motivations of our work to find some useful solutions for this problem. Our major contribution in this dissertation is to show the following.(1) To reveal the nature of SBL, explore the internal mechanism of its outstanding performance. Essential difference between the Type â… and the Type â…¡ maximum likelihood (empirical Bayes) used in Expectation Maximization Sparse Bayesian Learning (EMSBL) is discussed. Besides, we also reveal the relations between FOCUSS, IRLS and EMSBL. The local minimum property of EMSBL is demonstrated and compared with that of the L0 norm local solution via numerical simulation, which shows that, the former has less local minimum and better recovery effect than the latter. It means that EMSBL outperforms the other algorithms in the sense of mean square error and recovery efficiency.(2) To use a simple combination of existing algorithms to obtain excellent recovery capabilities, we propose a recovery algorithm SD_IRLp based on support drive for the purpose of recovering sparse high dimensional signals from few linear measurements, especially in the noisy case. The new method involves two steps as follows:(1)firstly, a relatively dense solution is found assuming there is no noise, and a support is pick-up by means of Thresholded Basis Pursuit (TBP), furthermore a weighting matrix as well as some parameters needed for next step are also computed; (2) with the detected support as a prior, a stationary solution is obtained via Iteratively Reweighted Lp Minimization (IRLp) procedure. Theoretic analysis and simulation results show that the proposed algorithm is a better compromise in the sense of computational complexity and recovery quality compared with 7 competitive methods similar as it. It is also proved that SD_IRLp is essentially a typical implement of a family of algorithms and can be extended expediently.(3) In order to overcome the weaknesses of the traditional SLO algorithm, we present an iteratively reweighted LO approximation framework using smooth differentiable surrogate functions. Basing on Newton direction, and consider the recovery procedure as a CCCP, two algorithms are also derived. All surrogate functions in literatures can be used in our framework. Extensive numerical simulations have demonstrated the robustness and applicability of our new theory and algorithms.(4) In order to use all types of prior information more effectively, after summarizing various types of prior information and their impacting on recovery algorithm, we research with focus on three ways to use prior information:(1) prior information is introduced as a weight for iterative computation; (2) inspired from compressed domain interference cancellation, a new algorithm called as Orthogonal Projection FOCUSS (OP_FOCUSS) is given, where the orthogonal projection technique is used to eliminate the influence of known supports; (3) as a generalization of OP_FOCUSS, a recovery framework, named Progressive Cancel FOCUSS (PC_FOCUSS) is also presented, here the detected elements can be canceled before each iteration, the related numerical experiments suggest that it can get notable performance improvement.(5) To enhance the switching efficiency of cognitive radio system, using Modulated Wideband Converter (MWC) as the sampling front end, a novel distributed wideband spectrum sensing scheme is proposed. During the spectrum sensing, information from neighbor nodes can be used as a priori, and a new algorithm named Information-Aided MSBL (IA-MSBL) with promising recovery performance is also introduced. As the simulation results shown, the proposed spectrum sensing system can effectively resist the interference and fading, and improve the spectrum sensing accuracy.Finally, the dissertation gives a full summarization and prospect. Moreover, it also prompts some open problems for future research.
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