Probabilistic inference beyond MAP estimation is of interest in computer vision, both for learning appropriate models and in applications. Yet, common approximate inference techniques, such as belief propagation, have largely been limited to discrete-valued Markov random fields (MRFs) and models with small cliques. Oftentimes, neither is desirable from an application standpoint. This paper studies mean field inference for continuous-valued MRF models with high-order cliques. Mean field can be applied effectively to such models by exploiting that the factors of certain classes of MRFs can be formulated using Gaussian mixtures, which allows retaining the mixture indicator as a latent variable. We use an image restoration setting to show that resulting mean field updates have a computational complexity quadratic in the clique size, which makes them scale even to large cliques. We contribute an empirical study with four applications: Image denoising, non-blind deblurring, noise estimation, and layer separation from a single image. We find mean field to yield a favorable combination of performance and efficiency, e.g. outperforming MAP estimation in denoising while being competitive with expensive sampling approaches. Novel approaches to noise estimation and layer separation demonstrate the breadth of applicability.
