Non-Local Retinex---A Unifying Framework and Beyond

In this paper, we provide a short review of Retinex and then present a unifying framework. The fundamental assumption of all Retinex models is that the observed image is a multiplication between the illumination and the true underlying reflectance of the object. Starting from Morel's 2010 PDE model, where illumination is supposed to vary smoothly and where the reflectance is thus recovered from a hard-thresholded Laplacian of the observed image in a Poisson equation, we define our unifying Retinex model in two similar, but more general, steps. We reinterpret the gradient thresholding model as variational models with sparsity constraints. First, we look for a filtered gradient that is the solution of an optimization problem consisting of two terms: a sparsity prior of the reflectance and a fidelity prior of the reflectance gradient to the observed image gradient. Second, since this filtered gradient almost certainly is not a consistent image gradient, we then fit an actual reflectance gradient to it, subject to further sparsity and fidelity priors. This generalized formulation allows making connections with other variational or kernel-based Retinex implementations. We provide simple algorithms for the optimization problems resulting from our framework. In particular, in the quadratic case, we can link our model to a plausible neural mechanism through Wilson--Cowan equations. Beyond unifying existing models, we derive entirely novel Retinex flavors by using more interesting non-local versions for the sparsity and fidelity priors. Eventually, we define within a single framework new Retinex applications to shadow detection and removal, nonuniformity correction, cartoon-texture decomposition, as well as color and hyperspectral image enhancement.