Variational Bayesian Approach and Gauss-Markov-Potts prior model

In many inverse problems such as 3D X-ray Computed Tomography (CT), the estimation of an unknown quantity, such as a volume or an image, can be greatly enhanced, compared to maximum-likelihood techniques, by incorporating a prior model on the quantity to reconstruct. A complex prior can be designed for multi-channel estimation such as reconstruction and segmentation thanks to Gauss-Markov-Potts prior model. For very large inverse problems such as 3D X-ray CT, maximization a posteriori (MAP) techniques are often used due to the huge size of the data and the unknown. Nevertheless, MAP estimation does not enable to have quantify uncertainties on the retrieved reconstruction, which can be useful for post-reconstruction processes for instance in industry and medicine. A way to tackle the problem of uncertainties estimation is to compute posterior mean (PM) for which the uncertainties are the variances of the posterior distribution. Because MCMC methods are not affordable for very large 3D problems, this paper presents an algorithm to jointly estimate the reconstruction and the uncertainties by computing PM thanks to variational Bayesian approach (VBA). The prior model we consider for the unknowns is a Gauss-Markov-Potts prior which has been shown to give good results in many inverse problems. After having detailed the used prior models, the algorithm based on VBA is detailed : it corresponds to an iterative computation ofapproximate distributions through the iterative updates of their parameters. The updating formulae are given in the last section. We also provide a method for initialization of the algorithm, as a method to fix each parameter. Perspectives are applications of this algorithm to large 3D problems such as 3D X-ray CT.