  This paper presents a scalable approximate Bayesian method for image
restoration using total variation (TV) priors. 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)
estimators 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.
