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Markov connected component fields

Part of: Stochastic processes Geometric probability and stochastic geometry Markov processes

Published online by Cambridge University Press:  01 July 2016

Jesper Møller  and
Rasmus Plenge Waagepetersen
Jesper Møller*
Affiliation:
Aalborg University
Rasmus Plenge Waagepetersen*
Affiliation:
Danish Institute of Agricultural Sciences
*
∗ Postal address: Department of Mathematics, Institute of Electronic Systems, Aalborg University, Fredrik Bajers Vej 7E, DK-9220 Aalborg, Denmark.
∗∗ Postal address: Department of Agricultural Systems, P.O. Box 50, Research Centre Foulum, DK-8830 Tjele, Denmark.
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Abstract

A new class of Gibbsian models with potentials associated with the connected components or homogeneous parts of images is introduced. For these models the neighbourhood of a pixel is not fixed as for Markov random fields, but is given by the components which are adjacent to the pixel. The relationship to Markov random fields and marked point processes is explored and spatial Markov properties are established. Extensions to infinite lattices are also studied, and statistical inference problems including geostatistical applications and statistical image analysis are discussed. Finally, simulation studies are presented which show that the models may be appropriate for a variety of interesting patterns, including images exhibiting intermediate degrees of spatial continuity and images of objects against background.

Keywords

Bayesian image analysisexponential family modelsgeostatisticsinfinite lattice processesMarkov chain Monte CarloMarkov partitionsMarkov random fieldsmaximum likelihoodnearest-neighbour Markov point processes

MSC classification

Primary: 60G60: Random fields
Secondary: 60D05: Geometric probability and stochastic geometry 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Stochatic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 30 , Issue 1 , March 1998 , pp. 1 - 35
Copyright
Copyright © Applied Probability Trust 1998 

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References

Aykroyd, R. G. and Green, P. J. (1991) Global and local priors, and the location of lesions using gamma camera imagery. Phil. Trans. R. Soc. London. A 337, 323–342.Google Scholar
Baddeley, A. J. and Möller, J. (1989) Nearest-neighbour Markov point processes and random sets. Int. Statist. Rev. 57, 89–121.CrossRefGoogle Scholar
Baddeley, A. J. and Van Lieshout, M. N. M. (1993) Stochastic geometry models in highlevel vision. In Statistics and Images, Advances in Applied Statistics, suppl. to J. Appl. Statist. 20, 235–256, eds Mardia, K.V. and Kanji, G.K., Carfax Publishing, Abingdon.Google Scholar
Baddeley, A. J., Kendall, W. S. and Van Lieshout, M. N. M. (1996a) Quermass-integral interaction processes. Warwick Statistics Research Report 293.Google Scholar
Baddeley, A. J., Van Lieshout, M. N. M. and Möller, J. (1996b) Markov properties of cluster processes. Adv. Appl. Prob. 28, 346–355.Google Scholar
Besag, J. E. (1975) Statistical analysis of non-lattice data. The Statistician 24, 179–195.Google Scholar
Besag, J. E. (1986) On the statistical analysis of dirty pictures. J. R. Statist. Soc. B 48, 259–302.Google Scholar
Besag, J. E., Milne, R. and Zachary, S. (1982) Point process limits of lattice processes. J. Appl. Prob. 19, 210–216.Google Scholar
Brodatz, P. (1966) Textures—a Photographic Album for Artists and Designers. Dover, New York.Google Scholar
Chayes, J. T., Chayes, L. and Kotecky, R. (1995) The analysis of the Widow–Rowlinson model by stochastic geometric methods. Commun. Math. Phys. 172, 551–569.Google Scholar
Clifford, P. (1990) Markov random fields in statistics. In Disorder in Physical Systems. A Volume in Honour of J. M. Hammersley. ed. Grimmett, G. R. and Welsh, D. J. A.. Clarendon Press, Oxford.Google Scholar
Comets, F. (1989) On consistency of a class of estimators for exponential families of Markov random fields on the lattice. Preprint. Université Paris-X.Google Scholar
Cormen, T. H., Leiserson, C. E. and Rivest, R. L. (1989) Introduction to Algorithms. McGraw-Hill, New York.Google Scholar
Cross, G. and Jain, A. (1983) Markov random field texture models. IEEE Trans. Pattern Analysis and Machine Intelligence 5, 25–39.CrossRefGoogle ScholarPubMed
Derin, H. and Elliot, H. (1987) Modelling and segmentation of noisy and textured images using Gibbs random fields. IEEE Trans. Pattern Analysis and Machine Intelligence 9, 39–55.Google Scholar
Fortuin, C. M. and Kasteleyn, P. W. (1972) On the random cluster model. I. Introduction and relation to other models. 57, 536–564.Google Scholar
Geman, S. and Geman, D. (1984) Stochastic Relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans. Pattern Analysis and Machine Intelligence 6, 721–741.Google Scholar
Geman, S. and Graffigne, C. (1986) Markov random fields and their applications to computer vision. In Proc. Int. Congress of Mathematics 1986. ed. Gleason, M.. American Mathematical Society, Providence, RI.Google Scholar
Georgii, H.-O. (1988) Gibbs Measures and Phase Transitions. Walter de Gruyter, Berlin.Google Scholar
Georgii, H.-O. and Häggström, O. (1996) Phase transition in continuum Potts models. Commun. Math. Phys. 181, 507–528.CrossRefGoogle Scholar
Geyer, C. J. (1994) On the convergence of Monte Carlo maximum likelihood calculations. J. R. Statist. Soc. B 56, 261–274.Google Scholar
Geyer, C. J. and Thompson, E. A. (1992) Constrained Monte Carlo maximum likelihood for dependent data. J. R. Statist. Soc. B 54, 657–699.Google Scholar
Gidas, B. (1988) Consistency of maximum likelihood and pseudolikelihood estimators for Gibbs distributions. In Stochastic Differential Systems, Stochastic Control Theory and Applications. IMA Volume 10, Springer, New York.Google Scholar
Green, P. J. (1986) Contribution to the discussion of the paper by J. Besag. J. R. Statist. Soc. B 48, 284–285.Google Scholar
Grenander, U. and Miller, M. (1994) Representations of knowledge in complex systems. J. R. Statist. Soc. B 56, 549–603.Google Scholar
Grimmett, G. (1995) The stochastic random-cluster process, and the uniqueness of random-cluster measures. Ann. Prob 23, 1461–1510.Google Scholar
Guyon, X. (1986) Estimation d'un champ de Gibbs. Preprint. Univ. Paris-I.Google Scholar
Häggström, O., van Lieshout, M. N. M. and Möller, J. (1996) Characterisation results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes. Technical Report R-96-2040, Department of Mathematics, Aalborg University. To appear in Bernoulli.Google Scholar
Haldorsen, H. H. and Damsleth, E. (1990) Stochastic modeling. J. Petroleum Tech. 404–412.Google Scholar
Hastings, W. (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109.Google Scholar
Helterbrand, J. D., Cressie, N. and Davidson, J. L. (1994) A statistical approach to identifying closed object boundaries in images. Adv. Appl. Prob. 26, 831–854.CrossRefGoogle Scholar
Hjort, N. L. and Omre, H. (1995) Topics in spatial statistics (with discussion). Scand. J. Statist. 21, 289–357.Google Scholar
Jacobsen, M. (1989) Existence and unicity of MLE in discrete exponential distributions. Scand. J. Statist. 16, 335–349.Google Scholar
Jensen, and Möller, J. (1991) Pseudolikelihood for exponential family models of spatial point processes. Ann. Appl. Prob. 3, 445–461.Google Scholar
Ji, C. and Seymour, L. (1996) Approximate Bayes model selection procedures for Gibbs–Markov fields. J. Statist. Planning Inference. Special Issue on Spatial Statistics, 51, 75–97.Google Scholar
Johnson, V. E. (1994) A model for segmentation and analysis of noisy images. J. Amer. Statist. Assoc. 89, 230–241.Google Scholar
Kendall, W. S. (1990) A spatial Markov property for nearest-neighbour Markov point processes. J. Appl. Prob. 28, 767–778.Google Scholar
Kindermann, R. and Snell, J. L. (1980) Markov Random Fields and their Applications. American Mathematical Society, Providence, RI.Google Scholar
Klein, W. (1982) Potts-model formulation of continuum percolation. Phys. Rev. B 26, 2677–2678.Google Scholar
Künsch, H., Geman, S. and Kehagias, A. (1995) Hidden Markov random fields. Ann. Appl. Prob. 5, 577–602.Google Scholar
Lebowitz, J. L. and Gallawotti, G. (1971) Phase transitions in binary lattice gases. J. Math. Phys. 12, 1129–1133.Google Scholar
Möller, J., (1994) Discussion contribution. Scand. J. Statist. 21, 346–349.Google Scholar
Möller, J., (1996) Markov chain Monte Carlo and spatial point processes. In Proc. Seminaire Européen de Statistique Toulouse. ed. Barndorff-Nielsen, O. E. et al., Chapman and Hall, London.Google Scholar
Preston, C. (1976) Random Fields (Lecture Notes in Mathematics 534) Springer, Berlin.Google Scholar
Ruelle, D. (1971) Existence of a phase transition in a continous classical system. Phys. Rev. Lett. 27, 1040–1041.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995) Stochastic Geometry and its Applications. Wiley, New York.Google Scholar
Sweeny, M. (1983) Monte Carlo study of weighted percolation clusters relevant to the Potts models. Phys. Rev. B 27, 4445–4455.Google Scholar
Swendsen, R. H. and Wang, J.-S. (1987) Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett. 58, 86–88.CrossRefGoogle ScholarPubMed
Syversveen, A.-R., and Omre, H. (1997) Conditioning of marked point processes with a Bayesian framework. Scand. J. Statist. 24, 341–352.CrossRefGoogle Scholar
Tjelmeland, H. and Besag, J. E. (1996) Markov random field models for objects against background. Technical report statistics 1/1996. Norwegian University of Science and Technology.Google Scholar
Tjelmeland, H. and Holden, L. (1993) Semi-Markov random fields. In Geostatistics Troia '92. Vol. 1. ed. Soares, A.. Kluwer, Amsterdam. pp. 479–492.Google Scholar
Waagepetersen, R. P. (1997) Phase transition and simulation for a penalised Ising model with applications in Bayesian image analysis. Research Report 381, Department of Theoretical Statistics, University of Aarhus.Google Scholar
Widow, B. and Rowlinson, J. S. (1970) New model for the study of liquid–vapor phase transitions. J. Chem. Phys. 52, 1670–1684.Google Scholar
Younes, L. (1988) Estimation and annealing for Gibbsian fields. Ann. Inst. Henri Poincaré 24, 269–294.Google Scholar