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Gibbs Field Approaches in Image Processing Problems


In this paper, we address the problem of image denoising using a stochastic differential equation approach. Proposed stochastic dynamics schemes are based on the property of diffusion dynamics to converge to a distribution on global minima of the energy function of the model, under a special cooling schedule (the annealing procedure). To derive algorithms for computer simulations, we consider discrete-time approximations of the stochastic differential equation. We study convergence of the corresponding Markov chains to the diffusion process. We give conditions for the ergodicity of the Euler approximation scheme. In the conclusion, we compare results of computer simulations using the diffusion dynamics algorithms and the standard Metropolis–Hasting algorithm. Results are shown on synthetic and real data.

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  1. 1.

    Geman, S. and Geman, D., Stochastic Relaxation, Gibbs Distribution, and the Bayesian Restoration of Images, IEEE Trans. Pattern Anal. Mach. Intell., 1984, vol. 6, no. 6, pp. 721–741.

    Google Scholar 

  2. 2.

    Geman, S. and Reynolds, G., Constrained Restoration and Recovery of Discontinuities, IEEE Trans. Pattern Anal. Mach. Intell., 1992, vol. 14, no. 3, pp. 367–383.

    Google Scholar 

  3. 3.

    Hajek, B., Cooling Schedules for Optimal Annealing, Math. Oper. Res., 1988, vol. 13, no. 2, pp. 311–329.

    Google Scholar 

  4. 4.

    Kirkpatrick, S., Gelatt, C.D., and Vecchi, M.P., Optimization by Simulated Annealing, Science, 1983, vol. 220, pp. 671–680.

    Google Scholar 

  5. 5.

    Holley, R. and Strook, D., Diffusions on an Inffinite Dimensional Torus, J. Funct. Anal., 1981, vol. 42, pp. 29–63.

    Google Scholar 

  6. 6.

    Albeverio, S., Kondratiev, Yu.G., and Roeckner, M., Uniqueness of the Stochastic Dynamics for Continuous Spin Systems on a Lattice, J. Funct. Anal., 1995, vol. 133, pp. 10–20.

    Google Scholar 

  7. 7.

    Geman, S. and Hwang, C., Diffusions for Global Optimization, SIAM J. Control Optim., 1986, vol. 24, no. 5, pp. 1031–1043.

    Google Scholar 

  8. 8.

    Chiang, T.-S., Hwang, C.-R., and Sheu, S.-J., Diffusion for Global Optimization in ℝn, SIAM J. Control Optim., 1987, vol. 25, no. 3, pp. 737–753.

    Google Scholar 

  9. 9.

    Winkler, G., Image Analysis, Random Fields, and Markov Chain Monte Carlo Methods: A Mathematical Introduction, New York: Springer, 2003.

    Google Scholar 

  10. 10.

    Freidlin, M.I. and Wentzell, A.D., Fluktuatsii v dinamicheskikh sistemakh pod deistviem malykh sluchainykh vozmushchenii, Moscow: Nauka, 1979. Translated under the title Random Perturbations of Dynamical Systems, Berlin: Springer, 1984.

    Google Scholar 

  11. 11.

    Wentzell, A.D., On the Asymptotic Behaviour of the Largest Eigenvalue of an Elliptic Second-Order Differential Operator with a Small Parameter by the Higher Derivatives, Dokl. kad. Nauk SSSR, 1972, vol. 202, no. 1, pp. 19–21.

    Google Scholar 

  12. 12.

    Wentzell, A.D., Formulas for Eigenfunctions and Eigenmeasures That Are Connected with a Markov Process, Theory Probab. Appl., 1973, vol. 18, no. 1, pp. 3–29.

    Google Scholar 

  13. 13.

    Liggett, T.M., Interacting Particle Systems, New York: Springer, 1985. Translated under the title Markovskie protsessy s lokal'nym vzaimodeistviem, Moscow: Mir, 1989.

    Google Scholar 

  14. 14.

    Ikeda, N. and Watanabe, S., Stochastic Differential Equations and Diffusion Processes, Amsterdam: North-Holland, 1981. Translated under the title Stokhasticheskie differentsial'nye uravneniya I diffuzionnye protsessy, Moscow: Nauka, 1986.

    Google Scholar 

  15. 15.

    Kloeden, P. and Platen, E., Numerical Solution of Stochastic Differential Equations, Berlin: Springer, 1992.

    Google Scholar 

  16. 16.

    Ethier, S.N. and Kurtz, T.G., Markov Processes: Characterization and Convergence, New York: Wiley, 1986.

    Google Scholar 

  17. 17.

    Dobrushin, R.L., Central Limit Theorem for Nonstationary Markov Chains. I, Theory Probab. Appl., 1956, vol. 1, no. 1, pp. 65–80.

    Google Scholar 

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Descombes, X., Zhizhina, E.A. Gibbs Field Approaches in Image Processing Problems. Problems of Information Transmission 40, 279–295 (2004).

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  • Stochastic Differential Equation
  • Equation Approach
  • Stochastic Dynamic
  • Image Denoising
  • Processing Problem