Partition function estimation of Gibbs random field images using Monte Carlo simulations

A Monte Carlo simulation technique for estimating the partition function of a general Gibbs random field image is proposed. By expressing the partition function as an expectation, an importance sampling approach for estimating it using Monte Carlo simulations is developed. As expected, the resulting estimators are unbiased and consistent. Computations can be performed iteratively by using simple Monte Carlo algorithms with remarkable success, as demonstrated by simulations. The work concentrates on binary, second-order Gibbs random fields defined on a rectangular lattice. However, the proposed methods can be easily extended to more general Gibbs random fields. Their potential contribution to optimal parameter estimation and hypothesis testing problems for general Gibbs random field images using a likelihood approach is anticipated. >

[1]  Anil K. Jain,et al.  Markov Random Field Texture Models , 1983, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[2]  M. Kac,et al.  A combinatorial solution of the two-dimensional Ising model , 1952 .

[3]  C. Geyer,et al.  Constrained Monte Carlo Maximum Likelihood for Dependent Data , 1992 .

[4]  Behnaam Aazhang,et al.  Constrained solutions in importance sampling via robust statistics , 1991, IEEE Trans. Inf. Theory.

[5]  D. K. Pickard.  Asymptotic inference for an Ising lattice III. Non-zero field and ferromagnetic states , 1979 .

[6]  J. W. Essam,et al.  Derivation of Low‐Temperature Expansions for the Ising Model of a Ferromagnet and an Antiferromagnet , 1965 .

[7]  A. Compagner,et al.  A special-purpose processor for the Monte Carlo simulation of ising spin systems , 1983 .

[8]  Wang,et al.  Nonuniversal critical dynamics in Monte Carlo simulations. , 1987, Physical review letters.

[9]  B. Gidas.  Nonstationary Markov chains and convergence of the annealing algorithm , 1985 .

[10]  T. Berger,et al.  Epsilon-entropy and Critical Distortion of Random Fields , 1990, IEEE Trans. Inf. Theory.

[11]  Richard E. Blahut,et al.  Principles and practice of information theory , 1987 .

[12]  Haluk Derin,et al.  Modeling and Segmentation of Noisy and Textured Images Using Gibbs Random Fields , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[13]  G. Potamianos,et al.  Stochastic Simulation Techniques for Partition Function Approximation of Gibbs Random Field Images , 1991 .

[14]  J. Besag.  On the Statistical Analysis of Dirty Pictures , 1986 .

[15]  Y. Ogata,et al.  Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure , 1981 .

[16]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[17]  John K. Goutsias,et al.  A theoretical analysis of Monte Carlo algorithms for the simulation of Gibbs random field images , 1991, IEEE Trans. Inf. Theory.

[18]  Mark Jerrum,et al.  Polynomial-Time Approximation Algorithms for Ising Model (Extended Abstract) , 1990, ICALP.

[19]  Anil K. Jain,et al.  Random field models in image analysis , 1989 .

[20]  PETER D. HORTENSIUS,et al.  Importance Sampling for Ising Computers Using One-Dimensional Cellular Automata , 1989, IEEE Trans. Computers.

[21]  John L. Richardson,et al.  A Fast Processor for Monte-Carlo Simulation , 1983 .

[22]  R. Baxter.  Exactly solved models in statistical mechanics , 1982 .

[23]  J. Besag.  Spatial Interaction and the Statistical Analysis of Lattice Systems , 1974 .

[24]  R. Ellis,et al.  Entropy, large deviations, and statistical mechanics , 1985 .

[25]  L. Onsager.  Crystal statistics. I. A two-dimensional model with an order-disorder transition , 1944 .

[26]  John K. Goutsias.  Unilateral approximation of Gibbs random field images , 1991, CVGIP Graph. Model. Image Process..

[27]  Y. Ogata,et al.  Likelihood Analysis of Spatial Point Patterns , 1984 .

[28]  Jerome Percus,et al.  Approximation Methods in Classical Statistical Mechanics , 1962 .

[29]  K. Wilson.  The renormalization group and critical phenomena , 1983 .

[30]  L. Younes.  Estimation and annealing for Gibbsian fields , 1988 .

[31]  John K. Goutsias,et al.  Mutually compatible Gibbs random fields , 1989, IEEE Trans. Inf. Theory.

[32]  John G. Kemeny,et al.  Finite Markov chains , 1960 .