Nonconvex Signal Reconstruction Models And Their Algorithms In Compressed Sensing

As an interdisciplinary and emerging sampling theory,compressed sensing mainly uses signal's sparsity characteristic to randomly sample the signal at a sampling rate far less than that required by the Nyquist sampling theorem,while recovering the original signal according to the related underdetermined linear equations and sparse signal reconstruction algorithms.It is extremely important how to construct appropriate sparse signal reconstruction models and explore their new optimization algorithm,which is of great significance with the rapid development of compressed sensing research.For single or multiple measurement vectors,nonconvex sparsity metrics,which characterize signal'sparsity,can more effectively approach l₀-or l_2,0-norm by comparison against convex sparse metrics.Additionally,the coherence property of a sensing matrix has important influence on the reconstruction of sparse signal under the same sparse metric.Therefore,in the study of sparse signal reconstruction,an important and fundamental topic is to design and study new nonconvex sparse metrics suitable for different types of sensing matrices and also to explore the theoretical foundations of the related nonconvex sparse reconstruction models.On the other hand,those traditional sparse signal reconstruction algorithms are difficult in seeking the sparse solutions of high-dimensional nonconvex reconstruction models,and thus it is important to probe into new optimization algorithms with high efficiency and the capability of parallel processing,in order to ensure that the problem of sparse signal reconstruction can be solved effectively.The current thesis concentrates on probing on the sparse signal reconstruction models and the related algorithms for exact or robust sparse signal reconstruction problems under single or multiple measurement vectors.In other words,as associated to the problem of nonconvex exact sparse signal reconstruction with a single measurement vector,the design and properties of new nonconvex sparse metrics are studied;meanwhile,the properties of solutions of the related reconstruction models are discussed.After that,new types of neural network models are designed and used for solving such reconstruction models.Their asymptotic behaviors are also discussed by means of the restricted isometry property or nonconvex smoothing approximation approaches.For the problem of robust nonconvex sparse reconstruction with single or multiple measurement vectors,a nonconvex sparse reconstruction model and a new Bayesian one are obtained by means of the penalty function method,nonconvex smoothing approximation techniques or Bayesian models.In order to handle such models,a new neural network is designed to solve the reconstruction model and its properties are studied as well.A sparse reconstruction algorithm,based on variational Bayesian inference is proposed to seek the sparsest solution of the new Bayesian model.The main work and innovations are summarized as follows.?1?To solve the problem of which the nonconvexity and nonsmoothness of the l₀-norm easily lead to the failure of sparse signal reconstruction,a single measurement vector-based exact sparse signal reconstruction model is established in terms of a separable nonconvex sparse metric.A smoothing approximation function and the Lipschitz property of one such nonconvex metric are obtained based on one smoothing approximation approach.Subsequently,a new smoothing recurrent neural network is proposed to find the sparse solution based on the KKT condition,in which its asymptotic behavior is discussed with the help of the nonsmooth analysis theory,differential equation theory and properties of the above smoothing approximation function.Comparative experiments validate that the TL1?Transformed l₁ function?measure-based separable nonconvex sparse metric not only is strongly robust to the sensing matrix,but also can well describe the sparsity of the signal.The proposed neural network with strong robustness to the coherence of the sensing matrix has clear advantages in recovering sparse signals.?2?Inspired by the fact that the nonconvex sparse metric l₁-l₂ is of satisfactory sparse signal reconstruction performance under high coherent sensing matrix,a sparse signal reconstruction model so-called l_1-p-p model,based on the difference of l₁-and l_p-norms?1<p?2?is constructed to formulate the problem of exact sparse signal reconstruction as related to a single measurement vector.The sufficient condition of which the model can accurately reconstruct the sparse solution is obtained by the restricted isometry property.Afterwards,a generalized gradient projection smoothing neural network,which can solve the l_1-p-p model in parallel,is proposed based on the inspirations of smoothing approximation and gradient projection.Its asymptotic properties are also discussed in terms of the nonsmooth analysis theory,differential equation theory and properties of smoothing function.By means of multiple types of algorithms,comparative experiments show that the solution of the l_1-p-p model has satisfactory sparsity under the low/high coherent sensing matrix and the small/large p.?3?For the problem of single measurement vector-based robust sparse signal reconstruction with Gaussian noise,a robust sparse signal reconstruction model is constructed,based on the TL1 sparse metric with fine reconstruction performance and strong robustness to the sensing matrix's coherence.As related to the penalty function method and a smooth approximation technique,a smoothing recurrent neural network is developed to seek the sparsest solution of the model,and meanwhile its asymptotic properties are found by the nonsmooth analysis theory,differential equation theory and properties of smoothing function.Numerically comparative experiments indicate that the obtained neural network can well reconstruct the sparse signal and is robust to Gaussian noise.?4?For the problem of multiple measurement vectors-based robust sparse signal reconstruction with impulsive noise,the multivariate generalized t-distribution and multivariate Laplacian distribution are used to characterize the impulsive noise prior and the prior of unknown joint sparse source signals,respectively.Then,a robust hierarchical Bayesian model is developed to describe the solution of the above sparse recovery problem.Generally,it is difficult to accurately obtain the posterior distributions of unknown parameters.In order to well approximate the posterior distributions so that the posterior mean-value estimations can be easily calculated,a multivariate generalized t-distribution variational Bayesian algorithm,based on variational Bayesian inference is proposed to infer the estimation of the source joint sparse signals and the posteriors of the hidden variables,by the variational Bayesian inference method.Numerically comparative experiments validate that the algorithm has distinct advantages over the compared methods and is strongly robust to different types of impulsive noises.