When we take a picture through transparent glass the image we
obtain is often a linear superposition of two images: the image of
the scene beyond the glass plus the image of the scene reflected
by the glass. Decomposing the single input image into two images
is a massively ill-posed problem: in the absence of additional
knowledge about the scene being viewed there are an infinite
number of valid decompositions. In this paper we focus on an
easier problem: user assisted separation in which the user
interactively labels a small number of gradients as belonging to
one of the layers.
Even given labels on part of the gradients, the problem is still ill-posed and additional prior knowledge is needed. Following recent results on the statistics of natural images we use a sparsity prior over derivative filters. We first approximate this sparse prior with a Laplacian prior and obtain a simple, convex optimization problem. We then use the solution with the Laplacian prior as an initialization for a simple, iterative optimization for the sparsity prior. Our results show that using a prior derived from the statistics of natural images gives a far superior performance compared to a Gaussian prior and it enables good separations from a small number of labeled gradients.