Sparse bayesian methods and their applications

The theory of compressed sensing (CS) has been extensively investigated and successfully applied in various areas over the past several decades. The key ingredient in this technique is the proper exploitation of sparsity, which allows the recovery of high-dimensional signals from their low-dimensional projections. Although a large number of sparse signal recovery algorithms have been proposed in the literature to achieve this objective, there still exist various opportunities and challenges in developing new algorithms to obtain more accurate and robust signal recovery performances. Among these algorithms, sparse Bayesian methods are recently developed to achieve higher accuracy and more flexibility. This thesis focuses on various issues of sparse Bayesian methods, including developing new algorithms and applying them to practical applications, such as radar imaging and wireless communication. In CS problems, the measurement matrix is often assumed to be known a priori, which unfortunately is not true in practical scenarios. The first task of the thesis is to consider the CS problems with multiplicative perturbations. We formulate this problem into a probabilistic model and develop an auto-calibration sparse Bayesian learning algorithm based on this model. In this method, signals and perturbations are iteratively estimated to achieve sparsity. Results from numerical experiments have demonstrated that the proposed algorithm achieves improvements on the accuracy of signal reconstruction. In radar imaging applications, phase errors often exist in the pre-processed data, which can be considered as a multiplicative perturbation model. However, different from the formulation in the first task, the received radar data are complex valued and the phase errors exhibit redundancy across range cells. To properly solve this problem, a multi-task Bayesian model is utilized to probabilistically model the sparse target scene and phase errors. The superiority of this method is that the uncertainty information of the estimation can be properly incorporated to obtain enhanced estimation accuracy. Experimental results based on synthetic and practical data have demonstrated that our method has a desirable de-noising capability and can produce a relatively well-focused image of the target, particularly in low signal-to-noise ratio (SNR) and high under-sampling ratio scenarios. As an extension of the sparse Bayesian auto-focus in the second task, we are motivated to further improve the imaging performances by incorporating structural information apart from sparsity. In this work, the structured sparse prior is imposed on the target scene in a statistical manner. Based on this statistical framework, the proposed algorithm can simultaneously cope with structured sparse recovery and phase error correction in an integrated manner. Due to the utilization of structured sparse constraint, the proposed algorithm can desirably preserve the target region and alleviate over-shrinkage problem, compared with the previous sparsity-driven auto-focus approaches. Moreover, to accelerate the convergence rate of the algorithm, we propose to adaptively eliminate portion of the noise-only range cells in the phase error estimation stage. The simulated and real data experimental results demonstrate that the proposed algorithm can obtain more concentrated imagery result with a much smaller number of iterations, particularly in low SNR and highly under-sampling scenarios. Finally, we consider a problem of recovering time-varying sparse signals with a particular structure. More specifically, the problem of estimating multiple frequency hopping signals with unknown hopping pattern is considered. Inspired by the sparse Bayesian learning algorithm, the problem is formulated hierarchically to induce sparsity. In addition to the sparsity, the hopping pattern is exploited via temporal-aware clustering by exerting a dependent Dirichlet process prior over the latent parametric space. The estimation accuracy of the parameters can be greatly improved by this particular information-sharing scheme, and sharp boundary of the hopping time estimation is manifested. Moreover, the proposed algorithm is further extended to multi-channel cases, where task-relation is utilized to obtain robust clustering of the latent parameters for better estimation performance. Since the problem is formulated in a full Bayesian framework, labor-intensive parameter tuning process can be avoided. Another superiority of the approach is that high-resolution instantaneous frequency estimation can be directly obtained without further refinement of the time frequency representation. Results of numerical experiments show that the proposed algorithm can achieve superior performance particularly in low SNR scenarios compared with other recently reported ones.Doctor of Philosophy (EEE