Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-20T23:33:23.215Z Has data issue: false hasContentIssue false
  • English
  • Français

Ising models and multiresolution quad-trees

Part of: Special processes Inference from stochastic processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  01 July 2016

W. S. Kendall  and
R. G. Wilson
W. S. Kendall*
Affiliation:
University of Warwick
R. G. Wilson*
Affiliation:
University of Warwick
*
∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: w.s.kendall@warwick.ac.uk
∗∗ Postal address: Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study percolation and Ising models defined on generalizations of quad-trees used in multiresolution image analysis. These can be viewed as trees for which each mother vertex has 2d daughter vertices, and for which daughter vertices are linked together in d-dimensional Euclidean configurations. Retention probabilities and interaction strengths differ according to whether the relevant bond is between mother and daughter or between neighbours. Bounds are established which locate phase transitions and show the existence of a coexistence phase for the percolation model. Results are extended to the corresponding Ising model using the Fortuin-Kasteleyn random-cluster representation.

Keywords

Gibbs stateIsing modelfinite island propertyimage analysisMarkov random fieldmultiresolution Markov random fieldpercolationquad-treerandom-cluster modelsegmentationunique infinite cluster

MSC classification

Primary: 62M40: Random fields; image analysis
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 96 - 122
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Benjamini, I. and Schramm, O. (1996). Percolation beyond Zd, many questions and a few answers. Electron. Commun. Prob. 1, No. 8, 71–82 (electronic).Google Scholar
Bouman, C. A. and Shapiro, M. (1994). A multiscale random field model for Bayesian image segmentation. IEEE Trans. Image Process. 3, 162–176.Google Scholar
Dobrushin, R. (1972). Coexistence of phase. Theory Prob. Appl. 17, 582–600.Google Scholar
Fortuin, C. (1972). On the random-cluster model. II. The percolation model. Physica 58, 393–418.Google Scholar
Fortuin, C. (1972). On the random-cluster model. III. The simple random-cluster model Physica 59, 545–570.CrossRefGoogle Scholar
Fortuin, C. and Kasteleyn, P. (1972). On the random-cluster model. I. Introduction and relation to other models. 57, 536–564.Google Scholar
Geman, D., Geman, S., Graffigne, C. and Dong, P. (1990). Boundary detection by constrained optimization. IEEE Trans. Pattern Anal. Mach. Intellig. 12, 609–628.Google Scholar
Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distribution, and Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intellig. 6, 721–741.Google Scholar
Georgii, H., Häggström, O. and Maes, C. (2001). The random geometry of equilibrium phases. In Phase Transitions and Critical Phenomena, eds Domb, C. and Lebowitz, J., Academic Press, London, pp. 1–142.Google Scholar
Gielis, G. and Grimmett, G. (2002). Rigidity of the interface in percolation and random-cluster models. J. Statist. Phys. 109, 1–37.Google Scholar
Gonzalez, R. and Woods, R. (2002). Digital Image Processing, 2nd edn. Prentice Hall, Upper Saddle River, NJ.Google Scholar
Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin.Google Scholar
Grimmett, G. and Newman, C. (1990). Percolation in ∞+1 dimensions. In Disorder in Physical Systems, eds Grimmett, G. and Welsh, D., Oxford University Press, pp. 167–190.Google Scholar
Häggström, O., Peres, Y. and Schonmann, R. (1999). Percolation on transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness. In Perplexing Problems in Probability, eds Bramson, M. and Durrett, R., Birkhäuser, Boston, MA, pp. 69–90.Google Scholar
Kato, Z., Berthod, M. and Zerubia, J. (1993). A hierarchical Markov random field model and multi-temperature annealing for parallel image classification. Tech. Rep. 1938, INRIA.Google Scholar
Laferte, J.-M., Perez, P. and Heitz, F. (2000). Discrete Markov image modeling and inference on the quadtree. IEEE Trans. Image Process. 9, 390–404.Google Scholar
Lavalle, S. M. and Hutchinson, S. A. (1995). A Bayesian segmentation methodology for parametric image models. IEEE Trans. Pattern Anal. Mach. Intellig. 17, 211–217.Google Scholar
Li, J., Gray, R. and Olshen, R. (2000). Multiresolution image classification by hierarchical modeling with two-dimensional hidden Markov models. IEEE Trans. Inf. Theory 46, 1826–1840.Google Scholar
Mallat, S. (1989). Multifrequency channel decompositions of images and wavelet models. IEEE Trans. Acoustics, Speech, Signal Process. 37, 2091–2110.Google Scholar
Manjunath, B. S. and Chellappa, R. (1991). Unsupervised texture segmentation using Markov random field models. IEEE Trans. Pattern Anal. Mach. Intellig. 13, 478–482.Google Scholar
Mignotte, M., Collet, C., Perez, P. and Bouthemy, P. (2000). Sonar image segmentation using an unsupervised hierarchical MRF model. IEEE Trans. Image Process. 9, 1216–1231.Google Scholar
Newman, C. and Wu, C. (1990). Markov fields on branching planes. Prob. Theory Relat. Fields 85, 539–552.Google Scholar
Panjwani, D. K. and Healey, G. (1995). Markov random field models for unsupervised segmentation of textured color images. IEEE Trans. Pattern Anal. Mach. Intellig. 17, 939–954.Google Scholar
Preston, C. (1977). Spatial birth-and-death processes. Bull. Inst. Internat. Statist. 46, 371–391.Google Scholar
Propp, J. and Wilson, D. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9, 223–252.Google Scholar
Schonmann, R. (2001). Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. Commun. Math. Phys. 219, 271–322.Google Scholar
Series, C. and Sinai, Y. (1990). Ising models on the Lobachevsky plane. Commun. Math. Phys. 128, 63–76.Google Scholar
Spitzer, F. (1975). Markov random fields on an infinite tree. Ann. Prob. 3, 387–398.CrossRefGoogle Scholar
Szeliski, R. (1989). Bayesian Modeling of Uncertainty in Low-level Vision. Kluwer, Dordrecht.Google Scholar
Wilson, R. and Li, C. (2002). A class of discrete multiresolution random fields and its application to image segmentation. To appear in IEEE Trans. Pattern Anal. Mach. Intellig. 25, 42–56.Google Scholar
Wilson, R. and Spann, M. (1988). Image Segmentation and Uncertainty. Research Studies Press, Letchworth.Google Scholar
Wu, C. (1996). Ising models on hyperbolic graphs. J. Statist. Phys. 85, 251–259.Google Scholar
Wu, C. (2000). Ising models on hyperbolic graphs. II. J. Statist. Phys. 100, 893–904.Google Scholar