Communications and Networks pp 266-298 | Cite as

# The Use of Gibbs Distributions In Image Processing

## Abstract

In this chapter, we discuss the potential of Gibbs distributions (GDs) as models in image processing applications. We briefly review the definitions and basic concepts of Markov random fields (MRFs) and GDs and present some realizations from these distributions. The use of statistical models and methods in image processing has increased considerably over the recent years. Most of these studies involve the use of MRF models and processing techniques based on these models. The pioneering work on MRFs due to Dobrushin [1], Wong [2], and Woods [3] involves extending the Markovian property in one dimension to higher dimensions. However, due to lack of causality in two dimensions the extension is not straightforward. Some properties in one dimension, for example, the equivalence of one-sided and two-sided Markovianity, do not carry over to two dimensions. The early work by Abend *et al*. [4] presents a causal characterization for a class of MRFs called Markov mesh random fields. This work also includes a formulation pointing out the Gibbsian property of a Markov chain without fully realizing the connection. Other attempts in extending the Markovian property to two dimensions include the autoregressive models: the “simultaneous autoregressive” (SAR) models and the “conditional Markov” models introduced by Chellappa and Kashyap [5].

## Keywords

Random Field Gray Level Joint Distribution Texture Image Markov Random Field## Preview

Unable to display preview. Download preview PDF.

## References

- [1]R. L. Dobrushin, The description of a random field by means of conditional probabilities and conditions of its regularity,
*Theory Prob. Appl.*, Vol. 13, pp. 197–224, 1968.CrossRefGoogle Scholar - [2]E. Wong, Recursive causal linear filtering for two-dimensional random fields,
*IEEE Trans. Inform. Theory*, Vol. IT-24, pp. 50 - 59, January 1978.Google Scholar - [3]J. W. Woods, Two-dimensional discrete Markovian fields,
*IEEE Trans. Inform. Theory*, Vol. IT-18, pp. 232–240, March 1972.Google Scholar - [4]K. Abend, T. J. Harley and L. N. Kanal, Classification of binary random patterns,
*IEEE Trans. Inform. Theory*, Vol. IT-11, pp. 538–544, October 1965.Google Scholar - [5]R. Chellappa and R. L. Kashyap, Digital image restoration using spatial interaction models,
*IEEE Trans. Acoust. Speech Sig. Proc.*, Vol. ASSP- 30, pp. 461–472, June 1982.CrossRefGoogle Scholar - [6]J. W. Woods and C. H. Radewan, Kalman filtering in two-dimensions,
*IEEE Trans. Inform. Theory*, Vol. IT-23, pp. 473–482, July 1977.Google Scholar - [7]F. R. Hansen and H. Elliott, Image segmentation using simple Markov field models,
*Comp. Graph. Im. Proc.*, Vol. 20, pp. 101–132, 1982.CrossRefGoogle Scholar - [8]H. Derin, H. Elliott, R. Cristi and D. Geman, Bayes smoothing algo¬rithms for segmentation of images modelled by Markov random fields,
*IEEE Trans. Pat. Analy. Mach. Intel.*, Vol. PAMI-6, pp. 707–720, November 1984.Google Scholar - [9]E. Ising,
*Zeitschrift Physik*, Vol. 31, p. 253, 1925.CrossRefGoogle Scholar - [10]D. Ruelle,
*Statistical Mechanics*, Benjamin, New York, 1977.zbMATHGoogle Scholar - [11]P. A. Flinn, Monte Carlo calculation of phase separation in a two- dimensional Ising system,
*Journ. Statist. Phys.*, Vol. 10, pp. 89–97, 1974.CrossRefGoogle Scholar - [12]J. Besag, Spatial interaction and the statistical analysis of lattice systems,
*Journ. Roy. Statist. Soc.*, series B., Vol. 36, pp. 192–226, 1974.MathSciNetzbMATHGoogle Scholar - [13]M. B. Averintzev, On a method of describing discrete parameter random fields,
*Problemy Peredacci Informacii*, Vol. 6, pp. 100–109, 1970.Google Scholar - [14]F. Spitzer, Markov random fields and Gibbs ensembles,
*Amer. Math. Mon.*, Vol. 78, pp. 142–154, February 1971.MathSciNetzbMATHCrossRefGoogle Scholar - [15]R. Kinderman and J. L. Snell,
*Markov Random Fields and Their Applications*, Providence, R.I.: American Mathematical Society, 1980.Google Scholar - [16]M. Hassner and J. Sklansky, The use of Markov random fields as models of texture,
*Comput. Graph. Im*. Proc., Vol. 12, pp. 357–370, 1980.CrossRefGoogle Scholar - [17]G. R. Cross and A. K. Jain, Markov random field texture models,
*IEEE Trans. Pat. Analy. Mach. Intel.*, Vol. PAMI-5, pp. 25–39, January 1983.Google Scholar - [18]18] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equation of state calculations by fast computing machines,
*Journ. Chem. Phys.*, Vol. 21, pp. 1087–1092, June 1953.CrossRefGoogle Scholar - [19]F. S. Cohen and D. B. Cooper, Real-time textured image segmentation based on non-causal Markovian random field models,
*Proc. of SPIE Conf. Intel Robot.*, Cambridge, MA., November 1983.Google Scholar - [20]S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, submitted to IEEE Trans. Pat. Analy. Mach. Intel., September1983.Google Scholar
- [21]H. Elliott, H. Derin, R. Cristi and D. Geman, Application of the Gibbs distribution to image segmentation, Proc. 1984
*IEEE Int. Conf. on ASSP*, San Diego, CA, March 1984.Google Scholar - [22]H. Elliott, H. Derin and R. Soucy, Modelling and segmentation of noisy and textured images using Gibbs random field models, Univ. Mass. Tech. Rep., Univerisity of Massachusetts, Amherst, MA, September 1984.Google Scholar
- 23] L. N. Kanal, Markov mesh models, in Image Modeling, New York, Academic Press, 1980.Google Scholar
- [24]P. A. P. Moran, A Gaussian Markovian process on a square lattice,
*J. Appl. Prob.*, Nol. 10, pp. 54–62, 1973.zbMATHCrossRefGoogle Scholar - [25]R. Chellappa and S. Chatterjee, Classification of textures using Gaussian Markov random fields, submitted to
*IEEE Trans. Acoust. Speech Sig. Proc*.Google Scholar - [26]H. Elliott and H. Derin, A new approach to parameter estimation for Gibbs random fields, submitted to 1985 Int. Conf. on Acoust. Speech Sig. Proc., August 1984.Google Scholar
- [27]R. Cristi, Analysis of some statistical properties for a class of spatial distributions: The Ising model with random parameters, Tech. Rep. Dept. Elect. Eng., The University of Michigan, Dearborn, May 1984.Google Scholar