Reconstruction of Gray-scale Images

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 Λ ⊂ Z2. This image is degraded by some noise, which is assumed to act independently in each site of Λ and to have the same distribution on all sites. For the estimator we use the mode of the posterior distribution: the so called 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). Experimental results and a detailed example are given in the text. We also provide a web page where additional information and examples can be found. Mathematics Subject Classification 1991: 62H11 62M40 68U10.