Hypermodels in the Bayesian imaging framework

We consider the problem of restoring an image from a noisy blurred copy of it with the additional qualitative information that the image contains sharp discontinuities whose sizes and locations are unknown. The flexibility of the Bayesian imaging framework is particularly convenient in the presence of such qualitative, rather than quantitative, information. By using a non-stationary Markov model with the variance of the innovation process also unknown, it is possible to take advantage of the qualitative prior information, and Bayesian techniques can be applied to estimate simultaneously the unknown and the prior variance. Here we present a unified approach to Bayesian signal processing and imaging, and show that with rather standard choices of hyperpriors we obtain some classical regularization methods, including the TV and the Perona-Malik regularization, as special cases. The application of Bayesian hyperprior models to imaging applications requires a careful organization of the computations to overcome the challenges coming from the large dimensionality. We explain how the computation of MAP estimates within the proposed Bayesian framework can be made very efficiently by a judicious use of Krylov iterative methods solutions and priorconditioners. The Bayesian approach, unlike deterministic estimation methods which produce a single solution image, provides a very natural way to assess the reliability of single image estimates by a Markov Chain Monte Carlo (MCMC) based analysis of the posterior. Computed examples illustrate the different features and the computational properties of the Bayesian hypermodel approach to imaging.