Electronic Journal of Statistics

On a nonparametric resampling scheme for Markov random fields

Lionel Truquet

Full-text: Open access


We study an extension to general Markov random fields of the resampling scheme given in Bickel and Levina (2006) [4] for texture synthesis with stationary Markov mesh models. The procedure generates bootstrap replicates of a sample using kernel regression and the principle of Gibbs sampling. Consistency of the bootstrap distribution is investigated under the Dobrushin contraction condition. Some simulation examples are given, in particular for the texture synthesis, for which the multiscale algorithm of Paget and Longstaff (1998) [27] is revisited.

Article information

Electron. J. Statist., Volume 5 (2011), 1503-1536.

First available in Project Euclid: 23 November 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M40: Random fields; image analysis
Secondary: 62G08: Nonparametric regression 62G09: Resampling methods

Random fields image analysis nonparametric regression resampling methods


Truquet, Lionel. On a nonparametric resampling scheme for Markov random fields. Electron. J. Statist. 5 (2011), 1503--1536. doi:10.1214/11-EJS644. https://projecteuclid.org/euclid.ejs/1322057435

Export citation


  • [1] Abend, K., Harley, T. J. and Kanal, L. N. (1965), Classification of binary random patterns IEEE Trans. Inform. Theory, IT-11, 538-544.
  • [2] Aillot, P., Monbet, V., Prevosto, M. (2007), Survey of stochastic models for wind data and sea state time series. Probabilistic Engineering Mechanics, 22, 113-126.
  • [3] Besag, J. (1974), Spatial interaction and the statistical analysis of lattice systems. J. R. Statis. Soc., Series B, Vol. 36, 192-236.
  • [4] Bickel, P., Levina, E. (2006), Texture synthesis and nonparametric resampling of random fields. The Annals of Statistics, Vol. 34, 4, 1751-1773.
  • [5] Champagnat, F. Idier, J. Goussard, Y. (1998), Stationary Markov random fields on a finite rectangular lattice. Information Theory, IEEE Transactions on, Vol. 44, Issue 7, 2901-2916
  • [6] Dachian, S. (1998), Nonparametric estimation for Gibbs random fields specified through one-point systems. Statistical Inference for Stochastic Processes, 1, 245-264.
  • [7] Diggle, P.J., Gates, D.J., Stibbard, A. (1987), A nonparametric estimator for pairwise-interaction point process. Biometrika, 74, 763-770.
  • [8] Dobrushin, R., L. (1970), Prescribing systems of random variables by conditional distributions. Theory of Probability and its Applications, 15(3), 458-486.
  • [9] Doukhan, P. (1994), Mixing: Properties and Examples. Lecture Notes in Statist. 85. Springer, New York.
  • [10] Doukhan, P., Louhichi S. (1999), A new weak dependence condition and applications to moment inequalities. Stoch. Proc. Appl. 84, 313-342.
  • [11] Doukhan P., Mayo N., Truquet L. (2009), Weak dependence, models and some applications. Metrika 69, 2-3, 199-225.
  • [12] Efros, A. A., Freeman, W. T. (2001), Image quilting for texture synthesis and transfer. In Proc. 28th Annual Conference on Computer Graphics and Interactive Techniques 341-346. ACM Press, New York.
  • [13] Efros, A. A., Leung, T. (1999), Texture synthesis by non-parametric sampling. In Proc. IEEE International Conference on Computer Vision 2, 1033-1038. IEEE Computer Soc., Washington.
  • [14] Ethier, S.N., Kurtz, T.Z. (1986), Markov processes: Characterization and convergence. Wiley, New York.
  • [15] Föllmer, H. (1988), Random fields and diffusion processes. Ecole d’Ete de Saint Flour XV XVII. Lecture Notes in Math. 1362 101 203. Springer, New York.
  • [16] Geman, S., Geman, D. (1984), Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell, 6, 721-741, 1984.
  • [17] Georgii, H.-O. (1988), Gibbs measures and phase transitions. de Gruyter, Berlin.
  • [18] Guyon, X. (1995), Random fields on a network. Springer, Berlin.
  • [19] Härdle, W., Vieu, P. (1992), Kernel regression smoothing of time series. JTSA, Vol 13, No. 3, 209-232.
  • [20] Künsch, H.R. (1989), The jackknife and the bootstrap for general stationary observations. Annals of Statistics, 17, 1217-1241.
  • [21] Kwatra, V., Essa, I., Bobick, A., Kwatra, N. (2005), Texture optimization for example-based synthesis. ACM Transactions on Graphics, SIGGRAPH 2005, August 2005.
  • [22] Liang, L., Liu, C., Xu, Y., Guo, B., Shum, H.-Y. (2001), Real-time texture synthesis by patch-based sampling. Technical Report MSR-TR-2001-40, Microsoft Research.
  • [23] Luettgen, M.R., Karl, W.C, Wilsky, A.S, Tenney, R.R. (1993), Multiscale Representations of Markov Random fields. IEEE transactions on signal processing, vol. 41, 12, 3377-3396.
  • [24] Meyn S.P., Tweedie R. (1993), Markov Chains and Stochastic Stability. 3rd edition, Springer, London.
  • [25] Monbet V., Marteau P.F. (2006), Local Grid Bootstrap for Stationary Markov Processes. J. Statistical Planning and Inference 136(10), 3319-3338.
  • [26] Paget, R. (1999), Nonparametric Markov random field models for natural texture images. PhD Thesis, University of Queensland.
  • [27] Paget, R., Longstaff, I. D. (1998), Texture synthesis via a noncausal nonparametric multiscale Markov random field IEEE Transactions on Image Processing, Vol. 7, 6, 925-931.
  • [28] Paparoditis, E., Politis, D. (2002), The local bootstrap for Markov processes. J. Statist. Plann. Inference, 108, 301-328.
  • [29] Pickard, D.K. (1980), Unilateral Markov fields. Adv. Appl. Proba., 12, 655-671.
  • [30] Politis, D., Romano, J. (1993), Nonparametric resampling for homogeneous strong mixing random fields. Journal of Multivariate Analysis, 47, 301-328.
  • [31] Rajarshi, M. (1990), Bootstrap in Markov sequences based on estimates of transition density. Ann. Inst. Math. Statist., 42, 253-268.
  • [32] Simon, B. (1979), A remark on Dobrushin’s uniqueness theorem. Comm. Math. Phys., 68, 183-185.