Journal of Applied Mathematics

Network flow optimization for restoration of images

Boris A. Zalesky

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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.

Article information

J. Appl. Math., Volume 2, Number 4 (2002), 199-218.

First available in Project Euclid: 30 March 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M40: Random fields; image analysis 90C10: Integer programming
Secondary: 90C35: Programming involving graphs or networks [See also 90C27] 68U10: Image processing


Zalesky, Boris A. Network flow optimization for restoration of images. J. Appl. Math. 2 (2002), no. 4, 199--218. doi:10.1155/S1110757X02110035.

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  • A. A. Ageev, Complexity of problems of minimization of polynomials in Boolean variables, Upravlyaemye Sistemy (1983), no. 23, 3–11 (Russian).
  • ––––, Minimization of quadratic polynomials of Boolean variables, Upravlyaemye Sistemy (1984), no. 25, 3–16 (Russian).
  • P. A. Ferrari, M. D. Gubitoso, and E. J. Neves, Reconstruction of gray-scale images, Methodol. Comput. Appl. Probab. 3 (2001), no. 3, 255–270.
  • L. R. Ford Jr. and D. R. Fulkerson, Flows in Networks, Princeton University Press, New Jersey, 1962.
  • D. Geman, Random fields and inverse problems in imaging, École d'été de Probabilités de Saint-Flour XVIII–-1988, Lecture Notes in Math., vol. 1427, Springer, Berlin, 1990, pp. 113–193.
  • S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intell. 6 (1984), 721–741.
  • B. Gidas, Metropolis-type Monte Carlo simulation algorithms and simulated annealing, Topics in Contemporary Probability and Its Applications, Probab. Stochastics Ser., CRC, Florida, 1995, pp. 159–232.
  • D. M. Greig, B. T. Porteous, and A. H. Seheult, Exact maximum a posteriori estimation for binary images, J. Roy. Statist. Soc. Ser. B 51 (1989), no. 2, 271–279.
  • C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, New Jersey, 1982.
  • J. C. Picard and H. D. Ratliff, Minimum cuts and related problems, Networks 5 (1975), no. 4, 357–370.