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Journal of Applied Mathematics
Volume 2, Issue 4, Pages 199-218

Network flow optimization for restoration of images

Institute of Engineering Cybernetics NAN, Surganov Street 6, Minsk 220012, Belarus

Received 5 October 2001

Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The network flow optimization approach is offered for restoration of gray-scale and color images corrupted by noise. The Ising models are used as a statistical background of the proposed method. We present the new multiresolution network flow minimum cut algorithm, which is especially efficient in identification of the maximum a posteriori (MAP) estimates of corrupted images. The algorithm is able to compute the MAP estimates of large-size images and can be used in a concurrent mode. We also consider the problem of integer minimization of two functions, U1(x)=λi|yixi|+i,jβi,j|xixj| and U2(x)=iλi(yixi)2+i,jβi,j(xixj)2, with parameters λ,λi,βi,j>0 and vectors x=(x1,,xn), y=(y1,,yn){0,,L1}n. Those functions constitute the energy ones for the Ising model of color and gray-scale images. In the case L=2, they coincide, determining the energy function of the Ising model of binary images, and their minimization becomes equivalent to the network flow minimum cut problem. The efficient integer minimization of U1(x),U2(x) by the network flow algorithms is described.