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Bayesian Estimation for Homogeneous and Inhomogeneous Gaussian Random Fields
May 1998 (vol. 20 no. 5)
pp. 533-539

Abstract—This paper investigates Bayesian estimation for Gaussian Markov random fields. In particular, a new class of compound model is proposed which describes the observed intensities using an inhomogeneous model and the degree of spatial variation described using a second random field. The coupled Markov random fields are used as prior distributions, and combined with Gaussian noise models to produce posterior distributions on which estimation is based. All model parameters are estimated, in a fully Bayesian setting, using the Metropolis-Hastings algorithm. The full posterior estimation procedures are illustrated and compared using various artificial examples. For these examples the inhomogeneous model performs very favorably when compared to the homogeneous model, allowing differential degrees of smoothing and varying local textures.

[1] G.R. Cross and A.K. Jain, "Markov Random Field Texture Models," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 5, no. 1, pp. 25-39, Jan. 1983.
[2] W. Qian and D.M. Titterington, "Multidimensional Markov Chain Models for Image Texture," J. Royal Statistical Soc. B, vol. 53, pp. 661-674, 1991.
[3] C.C. Cheng and C.L. Huang, "Markov Random-Fields for Texture Classification," Pattern Recognition Letters, vol. 14, pp. 907-914, 1993.
[4] R.G. Aykroyd, J.G.B. Haigh, and S. Zimeras, "Unexpected Spatial Patterns in Exponential Family Auto Models," Graphical Models and Image Processing, vol. 58, pp. 452-463, 1996.
[5] S. Geman and D.E. McClure, "Statistical Methods for Tomographic Image Reconstruction," Bull. Int. Statist. Inst., vol. 52, pp. 5-21, 1987.
[6] J. Besag, "Towards Bayesian Image Analysis," J. Appl. Statsist., vol. 16, pp. 395-407, 1989.
[7] P.J. Green, Bayesian Reconstruction from Emission Tomography Data Using a Modified Em Algorithm IEEE Trans. Medical Imaging, vol. 9, pp. 84-93, 1990.
[8] D. Geman and G. Reynolds, "Constrained Restoration and the Recovery of Discontinuities," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, pp. 367-383, 1992.
[9] S. Geman and D. Geman, "Stochastic Relaxation, Gibbs Distributions and the Bayesian Restoration of Images," IEEE Trans. Pattern Analysis and Machine Intelligence., vol. 6, pp. 721-741, 1984.
[10] F.C. Jeng and J.W. Woods, "Compound Gauss-Markov Random Fields for Image Estimation," IEEE Trans. Signal Processing, vol. 39, pp. 683-697, 1991.
[11] F.S. Cohen and D.B. Cooper, "Simple Parallel Hierarchical and Relaxation Algortihms for Segmenting Noncausal Markovian Random Fields," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 9, no. 2, pp. 195-219, Mar. 1987.
[12] H. Derin and H. Elliott, "Modeling and Segmentation of Noisy and Textured Images Using Gibbs Random Fields," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 9, no. 1, pp. 39-55, Jan. 1987.
[13] J.W. Woods, S. Dravida, and R. Mediavilla, "Image Estimation Using Doubly Stochastic Gaussian Random Field Models," IEEE Trans Pattern Analysis and Machine Intelligence, vol. 9, pp. 245-253, 1987.
[14] J. Besag, P.J. Green, D. Higdon, and K. Mengersen, "Bayesian Computation and Stochastic Systems," Statistical Science, vol. 10, pp. 1-41, 1995.
[15] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, "Equations of State Calculations by Fast Computing Machines," J. Chemical Physics, vol. 21, pp. 1,087-1,091, 1953.
[16] W.K. Hastings, "Monte Carlo Sampling Methods Using Markov Chains, and Their Applications," Biometrika, vol. 57, pp. 97-109, 1970.
[17] L.A. Shepp and Y. Vardi, "Maximum Likelihood Reconstruction in Positron Emission Tomography," IEEE Trans. Medical Imaging, vol. 1, pp. 113-122, 1982.
[18] J. Besag, "Spatial Interaction and the Statistical Analysis of Lattice Systems (With Discussion)," J. Royal Statistical Soc. B, vol. 36, pp. 192-236, 1974.
[19] J Besag and C Kooperberg, "On Conditional and Intrinisic AutoRegressions," Biometrika, vol. 82, pp. 733-746, 1995.
[20] J.O. Berger, Statistical Decision Theory and Bayesian Analysis, Springer Series in Statistics. New York: Spinger-Verlag, 1985.
[21] A.E. Raftery and J.D. Banfield, "Stopping the Gibbs Sampler, the Use of Morphology, and Other Issues in Spatial Statistics," Ann. Inst. Statist. Math., vol. 43, pp. 32-43, 1991.
[22] J. Besag, J.C. York, and A. Mollié, "Bayesian Image Restoration, With Two Applications in Spatial Statistics," Ann. Inst. Statist. Math., vol. 43, pp. 1-59, 1991.
[23] J.M. Hammersley and D.C. Hanscomb, Monte Carlo Methods, Methuen: London, 1964.
[24] D. Geman, "Random Fields and Inverse Problems in Imaging," Lecture Notes Math., vol. 1,427, pp. 113-193, 1991.
[25] P.J. Green and X.-L. Han, "Metropolis Methods, Gaussian Proposals, and Antithetic Variables," Lecture Notes in Statistics, vol. 74, pp. 142-164, 1992.
[26] J.G. Propp and B.D. Wilson, "Exact Sampling With Coupled Markov Chains and Applications to Statistical Mechanics," Random Structures and Algorithms, vol. 9, pp. 223-252, 1996.
[27] M.K. Cowles and B.P. Carlin, "Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review," J. Am. Statistical Soc., vol. 91, pp. 883-904, 1996.
[28] R.G. Aykroyd and P.J. Green, "Global and Local Priors, and the Location of Lesions Using Gamma-Camera Imagery," Phil. Trans. Royal Soc. London A, vol. 332, pp. 323-342, 1991.
[29] A.D. Sokal, "Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms," Cours de Troisième Cycle de la Physique en Suisse Romande, Lausanne, 1989.
[30] C.J. Geyer, "Estimating Normalizing Constants and Reweighted Mixtures in Markov Chain Monte Carlo," Tech. Rep. 568, School of Statistics, Univ. of Minnesota, 1994.

Index Terms:
Adaptive smoothing, compound Gauss-Markov random fields, coupled random fields, doubly stochastic random fields, Markov random fields, Metropolis-Hastings algorithm.
Citation:
Robert G. Aykroyd, "Bayesian Estimation for Homogeneous and Inhomogeneous Gaussian Random Fields," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 20, no. 5, pp. 533-539, May 1998, doi:10.1109/34.682182
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