  This paper presents a novel stochastic optimisation methodology to perform
empirical Bayesian inference in semi-blind image deconvolution problems. Given
a blurred image and a parametric class of possible operators, the proposed
optimisation approach automatically calibrates the parameters of the blur model
by maximum marginal likelihood estimation, followed by (non-blind) image
deconvolution by maximum-a-posteriori estimation conditionally to the estimated
model parameters. In addition to the blur model, the proposed approach also
automatically calibrates the noise variance as well as any regularisation
parameters. The marginal likelihood of the blur, noise variance, and
regularisation parameters is generally computationally intractable, as it
requires calculating several integrals over the entire solution space. Our
approach addresses this difficulty by using a stochastic approximation proximal
gradient optimisation scheme, which iteratively solves such integrals by using
a Moreau-Yosida regularised unadjusted Langevin Markov chain Monte Carlo
algorithm. This optimisation strategy can be easily and efficiently applied to
any model that is log-concave, and by using the same gradient and proximal
operators that are required to compute the maximum-a-posteriori solution by
convex optimisation. We provide convergence guarantees for the proposed
optimisation scheme under realistic and easily verifiable conditions and
subsequently demonstrate the effectiveness of the approach with a series of
deconvolution experiments and comparisons with alternative strategies from the
state of the art.
