Bayesian model selection for unsupervised image deconvolution with structured gaussian priors

This paper considers the objective comparison of stochastic models to solve inverse problems, more specifically image restoration. Most often, model comparison is addressed in a supervised manner, that can be time-consuming and partly arbitrary. Here we adopt an unsupervised Bayesian approach and objectively compare the models based on their posterior probabilities, directly from the data without ground truth available. The probabilities depend on the marginal likelihood or "evidence" of the models and we resort to the Chib approach including a Gibbs sampler. We focus on the family of Gaussian models with circulant covariances and unknown hyperparameters, and compare different types of covariance matrices for the image and noise.