Fast Scalable Image Restoration Using Total Variation Priors and Expectation Propagation

This paper presents a scalable approximate Bayesian method for image restoration using Total Variation (TV) priors, with the ability to offer uncertainty quantification. In contrast to most optimization methods based on maximum a posteriori estimation, we use the Expectation Propagation (EP) framework to approximate minimum mean squared error (MMSE) estimates and marginal (pixel-wise) variances, without resorting to Monte Carlo sampling. For the classical anisotropic TV-based prior, we also propose an iterative scheme to automatically adjust the regularization parameter via Expectation Maximization (EM). Using Gaussian approximating densities with diagonal covariance matrices, the resulting method allows highly parallelizable steps and can scale to large images for denoising, deconvolution, and compressive sensing (CS) problems. The simulation results illustrate that such EP methods can provide a posteriori estimates on par with those obtained via sampling methods but at a fraction of the computational cost. Moreover, EP does not exhibit strong underestimation of posteriori variances, in contrast to variational Bayes alternatives.