# Reconstruction of gray-scale images

## P. A. Ferrari, Marco D. Gubitoso, E. Jord\~ao Neves

We present an algorithm to reconstruct gray scale images corrupted by noise. We use a Bayesian approach. The unknown original image is assumed to be a realization of a Markov random field on a finite two dimensional region $\l \subset \zz$. This image is degraded by some noise, which is assumed to act independently in each site of $\l$ and to have the same distribution on all sites. For the estimator we use the mode of the posterior distribution: the so called {\it maximum a posteriori} (MAP) estimator. The algorithm, that can be used for both gray-scale and multicolor images, uses the binary decomposition of the intensity of each color and recovers each level of this decomposition using the identification of the problem of finding the two color MAP estimator with the min-cut max-flow problem in a binary graph, discovered by Greig, Porteous and Seheult (1989).