This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
A Simulation-Based Estimator for Hidden Markov Random Fields
August 1991 (vol. 13 no. 8)
pp. 825-830

An estimator for estimating the parameters of a Markov random field X from inaccurate observations is introduced. The author considers first a Markov (Gibbs) random field X=(X/sub i,j/) on a lattice L=((i,j): i=1,2,. . .,n; j=1,2,. . .,m). The marginal distributions of (X/sub i,j/,X/sub i+u,j+v/) (u,v=-1,0,1) are first estimated from an image. Then, random fields X* are simulated with the probability of X*/sub i+u,j+v/)=b nearly equal to the estimate of P(X/sub i,j/=X/sub i+u/,=b). A simulation method similar to the Gibbs sampler is used. The parameters of the Markov random field model are estimated from the X*'s with the pseudolikelihood method.

[1] M. P. Almeida and B. Gidas, "A variational method for estimating the parameters of MRF from complete or incomplete data," preprint, 1990.
[2] L. S. Anderson, "Consistent parameter estimation for corrupted Markov random fields with applications to image analysis," Dept. Stat. Univ. Washington, Rep. 117, 1988.
[3] J. Besag, "Statistical analysis of non-lattice data,"Statistician, vol. 24, pp. 179-195, 1975.
[4] J. Besag, "On the statistical analysis of dirty pictures,"J. Roy. Stat. Soc. Series B, vol. 48, pp. 259-302, 1986.
[5] B. Chalmond, "An iterative Gibbsian technique for reconstruction ofm-ary images,"Patt. Recog., vol. 22, pp. 747-761, 1989.
[6] F. Comets and B. Gidas,"Parameter estimation for Gibbs distributions from partially observed data," preprint, 1990.
[7] H. Derin and H. Elliott, "Modeling and segmentation of noisy and textured images using Gibbs random fields,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-9, pp. 39-55, Jan. 1987.
[8] S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-6, pp. 721-741, 1984.
[9] S. Geman and D. E. McClure, "Bayesian image analysis: An application to single photon emission tomography,"Proc. Amer. Stat. Assoc.: Statistical Computing Section, pp. 12-18, 1985.
[10] U. Grenander, "Tutorial in pattern theory," Division Appl. Math., Brown Univ., Providence, RI, 1983.
[11] U. Grenander, "Advances in pattern theory,"Ann. Stat., vol. 17, pp. 1-30, 1989.
[12] J. Haslett, "Maximum likelihood discriminant analysis on the plane using a Markovian model of spatial context,"Patt. Recog., vol. 18, pp. 287-296, 1985.
[13] N. L. Hjort, "Estimating parameters in neighborhood based classifiers for remotely sensed data, using unclassified vectors," inContextual Classification of Remotely Sensed Data Statistical Methods and Development of a System, H. V. Saebø, K. Bråten, N. L. Hjort, B. Llewellyn, and E. Mohn, Eds. Norwegian Computing Center, Rep. 768, 1985.
[14] M. D. Kirkland, "Simulation methods for Markov random fields," Ph. D. dissertation, Univ. Strathclyde, 1989.
[15] S. Lakshmanan and H. Derin, "Simultaneous parameter estimation and segmentation of Gibbs random fields using simulated annealing,"IEEE Trans. Patt. Anal. Machine Intell., vol. 11, pp. 799-813, 1989.
[16] J. Marroquin, S. Miter, and T. Poggio, "Probabilistic solution of ill-posed problems in computational vision,"J. Amer. Stat. Assoc., vol. 82, pp. 76-89, 1987.
[17] A. Penttinen, "Modelling interactions in spatial point patterns: Parameter estimation by the maximum likelihood method,"JyväskyläStudies Comput. Sci., Econ., Stat., vol. 7, 1984.
[18] W. Pieczynski, "Estimation of context in random fields,"Appl. Stat., vol. 16, pp. 283-290, 1989.
[19] A. Possolo, "Estimation of binary Markov random fields," Univ. Washington Tech. Rep., 1986.
[20] W. Qian and D. M. Titterington, "On the use of Gibbs Markov chain models in the analysis of images based on second-order pairwise interactive distributions,"Appl. Stat., vol. 16, pp. 267-281, 1989.
[21] B. D. R. Ripley,Stochastic Simulation. New York: Wiley, 1987.
[22] B. D. R. Ripley,Statistical Inference for Spatial Processes. London: Cambridge University Press, 1988.
[23] B. D. R. Ripley, "Statistics, images, and pattern recognition,"Canadian J. Stat., vol. 14, pp. 83-111, 1986.
[24] A. Veijanen, "On estimation of parameters of partially observed random fields and mixing processes," Finnish Statist. Soc., Helsinki, Tilastotieteellisiätutkimuksia 9, 1989.
[25] L. Younes, "Parametric inference for imperfectly observed Gibbsian fields,"Probab. Theory Related Fields, vol. 82, pp. 625-645, 1989.

Index Terms:
parameter estimation; picture processing; simulation-based estimator; hidden Markov random fields; probability; Gibbs sampler; pseudolikelihood method; Markov processes; parameter estimation; picture processing; probability
Citation:
A. Veijanen, "A Simulation-Based Estimator for Hidden Markov Random Fields," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, no. 8, pp. 825-830, Aug. 1991, doi:10.1109/34.85674
Usage of this product signifies your acceptance of the Terms of Use.