A relaxed factorial Markov random field for colour and depth estimation from a single foggy image

In this paper, we present a method to recover the albedo and depth from a single image. To this end, we depart from the scattering theory in the atmospheric vision model used elsewhere for defogging and dehazing. We then view the image as a relaxed factorial Markov random field (FMRF) of albedo and depth layers. This leads to a formulation which, for each of the layers in the FMRF, is akin to relaxation labelling problems. Moreover, we can obtain sparse representations for the graph Laplacian and Hessian matrices involved. This implies that global minima for each of the layers can be estimated efficiently via sparse Cholesky factorisation methods. We illustrate the utility of our method for depth and albedo recovery making use of real world data and compare against other techniques elsewhere in the literature.