Bayesian Compressive Sensing: Theory And Algorithm

In this thesis, we mainly concentrate on two problems, which are one bit compressive sensing and support knowledge aided Bayesian compressive sensing.Firstly, we consider the problem of sparse signal recovery from one-bit measurements. Due to the noise present in the acquisition and transmission process, some quantized bits may be flipped to their opposite states. These bit-flip errors, also referred to as the sign-flip errors, may result in severe performance degradation. To address this issue,we introduce a robust Bayesian compressed sensing framework to account for sign flip errors. Specifically, sign-flip errors are considered as a result of a sparse noise corrupted model, in which original(unquantized) observations are corrupted by sparse(impulse)noise. Numerical results are provided to illustrate the effectiveness and superiority of the proposed method.Secondly, it has been shown both experimentally and theoretically that sparse signal recovery can be significantly improved given that part of the signal’s support is known a priori. In practice, however, such prior knowledge is usually inaccurate and contains errors. Using such knowledge may result in severe performance degradation or even recovery failure. Based on the conventional sparse Bayesian learning framework, we propose a modified two-layer Gaussian-inverse Gamma hierarchical prior model and, moreover, an improved three-layer hierarchical prior model. The modified two-layer model employs an individual parameter for each sparsity controlling hyper parameter , and has the ability to place nonsparsity-encouraging priors to those coefficients that are believed in the support set. The three-layer hierarchical model is built on the modified two-layer prior model, with a prior placed on the parameters in the third layer. Such a model enables to automatically learn the true support from partly erroneous information through learning the values of the parameters . Variational Bayesian algorithms are developed based on the proposed hierarchical prior models. Numerical results are provided to illustrate the performance of the proposed algorithms.