Two-Dimensional Pattern-Coupled Sparse Bayesian Learning via Generalized Approximate Message Passing

We consider the problem of recovering two-dimensional (2-D) block-sparse signals with \emph{unknown} cluster patterns. Two-dimensional block-sparse patterns arise naturally in many practical applications such as foreground detection and inverse synthetic aperture radar imaging. To exploit the block-sparse structure, we introduce a 2-D pattern-coupled hierarchical Gaussian prior model to characterize the statistical pattern dependencies among neighboring coefficients. Unlike the conventional hierarchical Gaussian prior model where each coefficient is associated independently with a unique hyperparameter, the pattern-coupled prior for each coefficient not only involves its own hyperparameter, but also its immediate neighboring hyperparameters. Thus the sparsity patterns of neighboring coefficients are related to each other and the hierarchical model has the potential to encourage 2-D structured-sparse solutions. An expectation-maximization (EM) strategy is employed to obtain the maximum a posterior (MAP) estimate of the hyperparameters, along with the posterior distribution of the sparse signal. In addition, the generalized approximate message passing (GAMP) algorithm is embedded into the EM framework to efficiently compute an approximation of the posterior distribution of hidden variables, which results in a significant reduction in computational complexity. Numerical results are provided to illustrate the effectiveness of the proposed algorithm.