J. Appl. Math. 2 (4), 199-218, (21 August 2002) DOI: 10.1155/S1110757X02110035
KEYWORDS: 62M40, 90C10, 90C35, 68U10

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, ${U}_{1}\left(\mathbf{x}\right)=\lambda {\sum}_{i}\left|{y}_{i}-{x}_{i}\right|+{\sum}_{i,j}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta}_{i,j}\left|{x}_{i}-{x}_{j}\right|$ and ${U}_{2}\left(\mathbf{x}\right)={\sum}_{i}\text{\hspace{0.17em}}{\lambda}_{i}\text{\hspace{0.17em}}{\left({y}_{i}-{x}_{i}\right)}^{2}+{\sum}_{i,j}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta}_{i,j}\text{\hspace{0.17em}}{\left({x}_{i}-{x}_{j}\right)}^{2}$, with parameters $\lambda ,{\lambda}_{i},{\beta}_{i,j}>0$ and vectors $\mathbf{x}=\left({x}_{1},\dots ,{x}_{n}\right)$, $\mathbf{y}=\left({y}_{1},\dots ,{y}_{n}\right)\in {\left\{0,\dots ,L-1\right\}}^{n}$. 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 ${U}_{1}\left(\mathbf{x}\right),{U}_{2}\left(\mathbf{x}\right)$ by the network flow algorithms is described.