Bayesian approach with prior models which enforce sparsity in signal and image processing

In this review article, we propose to use the Bayesian inference approach for inverse problems in signal and image processing, where we want to infer on sparse signals or images. The sparsity may be directly on the original space or in a transformed space. Here, we consider it directly on the original space (impulsive signals). To enforce the sparsity, we consider the probabilistic models and try to give an exhaustive list of such prior models and try to classify them. These models are either heavy tailed (generalized Gaussian, symmetric Weibull, Student-t or Cauchy, elastic net, generalized hyperbolic and Dirichlet) or mixture models (mixture of Gaussians, Bernoulli-Gaussian, Bernoulli-Gamma, mixture of translated Gaussians, mixture of multinomial, etc.). Depending on the prior model selected, the Bayesian computations (optimization for the joint maximum a posteriori (MAP) estimate or MCMC or variational Bayes approximations (VBA) for posterior means (PM) or complete density estimation) may become more complex. We propose these models, discuss on different possible Bayesian estimators, drive the corresponding appropriate algorithms, and discuss on their corresponding relative complexities and performances.