Stochastic Alternating-Direction Expectation Propagation for High-Dimensional Inverse Problems in Imaging

It is a challenging task to reconstruct images from down-sampled and noisy measurements, such as Magnetic Resonance Imaging (MRI) and low-dose Computed Tomography (CT), due to the inherent mathematical ill-posedness of the inverse problem. This paper makes two contributions to Bayesian image reconstruction. Firstly, we propose stochastic alternating-direction expectation propagation (SAEP), a novel alternative to expectation propagation (EP), a family of variational inference algorithms. SAEP addresses the instability problem within EP due to negative variance. Secondly, we present SAEP as a black-box variational algorithm that leverages the Monte Carlo sampler to estimate the moments of the EP tilted distributions. SAEP allows scalable and robust Bayesian image reconstruction in cases where the tilted posterior is a mixture of different (unmatched) distributions that renders direct moment matching in Kullback-Leibler (KL) divergence infeasible. Further, as opposed to EP, which has no guarantee of convergence, SAEP can be shown to be convergent. We compared the performance of our SAEP algorithm to standard Markov Chain Monte Carlo (MCMC) on Gamma-camera imaging experiments and show that our SAEP algorithm outperforms MCMC in computing time while providing similar parameter estimates.