Mutually compatible Gibbs random fields

A unified theory for the mathematical description of Gibbs random fields that answers some important theoretical and practical questions about their statistical behavior is presented. The local transfer function is introduced, and the joint probability measure of the general Gibbs random field is derived in terms of this function. The resulting probability structure is required to satisfy the property of mutual compatibility. A necessary and sufficient condition for a Gibbs random field to be mutually compatible is developed and used to prove that a mutually compatible Gibbs random field is a unilateral Markov random field. The existence of some special nontrivial cases of Gibbs random fields that are mutually compatible is demonstrated. Conditions on the translation invariance and isotropy of the general Gibbs random field with a free boundary are studied. The class of Gibbs random fields with a homogeneous local transfer function and the class of horizontally and vertically translation-invariant Gibbs random fields are introduced and treated. The concept of a translation-invariant Gibbs random field is also explored. The problem of the statistical inference of mutually compatible Gibbs random fields is discussed. >

[1]  John Goutsias.  Mutually compatible Gibbs images: properties, simulation and identification , 1988, ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing.

[2]  Rama Chellappa,et al.  Classification of textures using Gaussian Markov random fields , 1985, IEEE Trans. Acoust. Speech Signal Process..

[3]  D. K. Pickard.  A curious binary lattice process , 1977, Journal of Applied Probability.

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

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

[6]  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.

[7]  Fernand S. Cohen,et al.  Markov random fields for image modelling and analysis , 1986 .

[8]  F. Spitzer.  Markov Random Fields and Gibbs Ensembles , 1971 .

[9]  John P. Moussouris.  Gibbs and Markov random systems with constraints , 1974 .

[10]  Laveen N. Kanal,et al.  Markov mesh models , 1980 .

[11]  David K. Pickard.  Unilateral Ising Models , 1978 .

[12]  D. K. Pickard.  Inference for Discrete Markov Fields: The Simplest Nontrivial Case , 1987 .

[13]  Laveen N. Kanal,et al.  Classification of binary random patterns , 1965, IEEE Trans. Inf. Theory.

[14]  W. Weidlich,et al.  THE STATISTICAL DESCRIPTION OF POLARIZATION PHENOMENA IN SOCIETY , 1971 .

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

[16]  Rama Chellappa,et al.  Estimation and choice of neighbors in spatial-interaction models of images , 1983, IEEE Trans. Inf. Theory.

[17]  Donald Geman,et al.  Bayes Smoothing Algorithms for Segmentation of Binary Images Modeled by Markov Random Fields , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[18]  D. K. Pickard,et al.  Unilateral Markov fields , 1980, Advances in Applied Probability.