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Tree Approximations to Markov Random Fields
April 1995 (vol. 17 no. 4)
pp. 391-402

Abstract—Methods for approximately computing the marginal probability mass functions and means of a Markov random field (MRF) by approximating the lattice by a tree are described. Applied to the a posteriori MRF these methods solve Bayesian spatial pattern classification and image restoration problems. The methods are described, several theoretical results concerning fixed-point problems are proven, and four numerical examples are presented, including comparison with optimal estimators and the Iterated Conditional Mode estimator and including two agricultural optical remote sensing problems.

[1] J. Besag,“Spatial interaction and the statistical analysis of lattice systems,” J. Royal Stat. Soc. B, vol. 36, no. 2, pp. 192-236, 1974.
[2] J. Besag,“On the statistical analysis of dirty pictures,” J. Royal Stat. Soc. B, vol. 48, pp. 259-302, 1986.
[3] S. Geman and D. Geman,“Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Transactions on PAMI, vol. 6, no. 6, pp. 721-741, Nov. 1984.
[4] D. Geman,S. Geman,C. Graffigne,, and P. Dong,“Boundary detection by constrained optimization,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 609-628, July 1990.
[5] D.M. Greig,B.T. Porteous,, and A.H. Seheult,“Exact maximum a posteriori estimation for binary images,” J. Royal Stat. Soc. B, vol. 51, no. 2, pp. 271-279, 1989.
[6] J. Marroquin,S. Mitter,, and T. Poggio,“Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc., vol. 82, no. 397, pp. 76-89, 1987.
[7] H.P. Hiriyannaiah,G.L. Bilbro,W.E. Snyder,, and R.C. Mann,“Restoration of piecewise-constant images by mean-field annealing,” J. Opt. Soc. Am. A, vol. 6, no. 12, pp. 1901-1912, 1989.
[8] D. Geiger and F. Girosi,“Parallel and deterministic algorithms from MRFs: Surface reconstruction,” IEEE Transactions on PAMI, vol. 13, no. 5, pp. 401-412, May 1991.
[9] J. Zhang,“The mean field theory in EM procedures for Markov random fields,” IEEE Transactions on Sig. Proc., vol. 40, no. 10, pp. 2570-2583, Oct. 1992.
[10] J. Zhang,“The mean field theory in EM procedures for blind Markov random field image restoration,” IEEE Transactions on Image Proc., vol. 2, no. 1, pp. 27-40, Jan. 1993.
[11] L. Herault and R. Horaud, “Figure Ground Discrimination: A Combinatorial Optimization Method,” IEEE Tran. Pattern Analysis and Machine Intelligence, vol. 15, no. 9, pp. 899-914, Sept. 1993.
[12] I.M. Elfadel,“From random fields to networks,” RLE Technical Report 579, The Research Laboratory of Electronics, MIT, Cambridge, MA, June 1993.
[13] C.-h. Wu and P.C. Doerschuk,“Cluster expansions for the deterministic computation of Bayesian estimators based on Markov random fields,” IEEE Transactions on PAMI, vol. 17, no. 3, pp. 275-293, March 1995.
[14] C.-h. Wu,Deterministic Parallelizable Solutions for Bayesian Markov Random Field Estimation Problems, PhD thesis, Purdue University, West Lafayette, IN, USA, May 1994.
[15] H. Derin,H. Elliott,R. Cristi,, and D. Geman,“Bayes smoothing algorithms for segmentation of binary images modeled by Markov random fields,” IEEE Transactions on PAMI, vol. 6, no. 6, pp. 707-720, 1984.
[16] P.A. Devijver,“Segmentation of binary images using third-order Markov mesh image models,” in Proc. 8th Internat. Conf. Pattern Recognition, Oct. 1986, pp. 259-261,Paris, France.
[17] P.A. Devijver and M.M. Dekesel,“Learning the parameters of a hidden Markov random field image model: A simple example,” in Pattern Recognition Theory and Applications, P.A. Devijver and J. Kittler, Eds., pp. 141-163. Springer-Verlag, 1987.
[18] R.J. Baxter,Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982.
[19] R. Kindermann and J.L. Snell,Markov Random Fields and Their Applications, American Mathematical Society, Providence, RI, 1980.
[20] J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, Inc., San Diego, CA, 1970.
[21] P.J.M. Laarhoven and E.H.L. Aarts, Simulated Annealing: Theory and Applications. D. Reidel Publishing, 1987.
[22] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes in C.Cambridge, England: Cambridge Univ. Press, 1988.
[23] B. Jeon and D.A. Landgrebe,“Classification with spatio-temporal interpixel class dependency contexts,” IEEE Transactions on Geosci. Remote Sensing, vol. 30, no. 4, pp. 663-672, July 1992.
[24] D.A. Landgrebe,“The development of a spectral-spatial classifier for earth observational data,” Pattern Recognition, vol. 12, pp. 165-175, 1980.
[25] L.E. Blanchard and O. Weinstein,“Design challenges of the thematic mapper,” IEEE Transactions on Geosci. Remote Sensing, vol. 18, no. 2, pp. 146-160, Apr. 1980.

Index Terms:
Markov random field, Bayesian estimation, spatial pattern classification, image segmentation, image restoration.
Citation:
Chi-hsin Wu, Peter C. Doerschuk, "Tree Approximations to Markov Random Fields," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 17, no. 4, pp. 391-402, April 1995, doi:10.1109/34.385979
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