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.

翻译：本文提出了一种新的随机优化方法，用于在半盲图像去卷积问题中执行经验贝叶斯推断。给定模糊图像和参数化的可能算子类别，所提出的优化方法通过最大边际似然估计自动校准模糊模型的参数，随后在条件于估计模型参数的情况下，通过最大后验估计进行（非盲）图像去卷积。除模糊模型外，该方法还自动校准噪声方差以及任何正则化参数。模糊、噪声方差和正则化参数的边际似然通常在计算上难以处理，因为它需要计算整个解空间上的多个积分。我们的方法通过使用随机近似近端梯度优化方案来解决这一难题，该方案利用Moreau-Yosida正则化未调整朗之万马尔可夫链蒙特卡罗算法迭代求解此类积分。这种优化策略可以轻松高效地应用于任何对数凹模型，并且只需使用通过凸优化计算最大后验解所需的相同梯度和近端算子。我们在现实且易于验证的条件下为所提出的优化方案提供了收敛保证，随后通过一系列去卷积实验以及与现有替代策略的比较，展示了该方法的有效性。