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Bayesian image restoration, with two applications in spatial statistics

  • Bayesian Image Analysis (with Discussion)
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Abstract

There has been much recent interest in Bayesian image analysis, including such topics as removal of blur and noise, detection of object boundaries, classification of textures, and reconstruction of two- or three-dimensional scenes from noisy lower-dimensional views. Perhaps the most straightforward task is that of image restoration, though it is often suggested that this is an area of relatively minor practical importance. The present paper argues the contrary, since many problems in the analysis of spatial data can be interpreted as problems of image restoration. Furthermore, the amounts of data involved allow routine use of computer intensive methods, such as the Gibbs sampler, that are not yet practicable for conventional images. Two examples are given, one in archeology, the other in epidemiology. These are preceded by a partial review of pixel-based Bayesian image analysis.

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References

  • Besag, J. E. (1974). Spatial interaction and the statistical analysis of lattice systems (with Discussion), J. Roy. Statist. Soc. Ser. B, 36, 192–236.

    Google Scholar 

  • Besag, J. E. (1975). Statistical analysis of non-lattice data, The Statistician, 24, 179–195.

    Google Scholar 

  • Besag, J. E. (1983). Discussion of paper by P. Switzer, Bull. Internat. Statist. Inst., 50 (Bk. 3), 422–425.

    Google Scholar 

  • Besag, J. E. (1986). On the statistical analysis of dirty pictures (with Discussion), J. Roy. Statist. Soc. Ser. B, 48, 259–302.

    Google Scholar 

  • Besag, J. E. (1989). Towards Bayesian image analysis, Journal of Applied Statistics, 16, 395–407.

    Google Scholar 

  • Besag, J. E. and Mollié, A. (1989). Bayesian mapping of mortality rates, Bull. Internat. Statist. Inst., 53 (Bk. 1), 127–128.

    Google Scholar 

  • Breslow, N. E. (1984). Extra-Poisson variation in log-linear models, J. Roy. Statist. Soc. Ser. C, 33, 38–44.

    Google Scholar 

  • Buck, C. E., Cavanagh, W. G. and Litton, C. D. (1988). The spatial analysis of soil phosphate data, Tech. Report, Department of Mathematics, University of Nottingham, U.K.

    Google Scholar 

  • Chow, Y., Grenander, U. and Keenan, D. M. (1988). Hands: a pattern theoretic study of biological shape, Tech. Report, Division of Applied Mathematics, Brown University, Providence, Rhode Island.

    Google Scholar 

  • Clayton, D. and Kaldor, J. (1987). Empirical Bayes estimates of age-standardized relative risks for use in disease mapping, Biometrics, 43, 671–681.

    Google Scholar 

  • Geman, D. and Geman, S. (1986). Bayesian image analysis, Disorderd Systems and Biological Organization (eds. E. Bienenstock et al.), in NATO ASI Series, Vol. F20, Springer, Berlin.

    Google Scholar 

  • Geman, D., Geman, S., Graffigne, C. and Ping, Dong (1990). Boundary detection by constrained optimization, I.E.E.E. Transactions: Pattern Analysis and Machine Intelligence, 12, 609–628.

    Google Scholar 

  • Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, I.E.E.E. Transactions: Pattern Analysis and Machine Intelligence, 6, 721–741.

    Google Scholar 

  • Geman, S. and Graffigne, C. (1987). Markov random field image models and their applications to computer vision, Proc. International Congress of Mathematicians (1986) (ed. A. M. Gleason), 1496–1517, Berkeley, California.

  • Geman, S. and McClure, D. (1987). Statistical methods for tomographic image reconstruction, Bull. Internat. Statist. Inst., 52 (Bk. 4), 5–21.

    Google Scholar 

  • Green, P. J. (1990). Penalized likelihood reconstructions from emission tomography data using a modified EM algorithm, I.E.E.E. Transactions: Medical Imaging, 9, 84–93.

    Google Scholar 

  • Greig, D. M., Porteous, B. T. and Seheult, A. H. (1989). Exact maximum a posteriori estimation for binary images, J. Roy. Statist. Soc. Ser. B, 51, 271–279.

    Google Scholar 

  • Grenander, U. (1983). Tutorial in pattern theory, Tech. Report, Division of Applied Mathematics, Brown University, Providence, Rhode Island.

    Google Scholar 

  • Kent, J. T. and Mardia, K. V. (1988). Spatial classification using fuzzy membership models, I.E.E.E. Transactions: Pattern Analysis and Machine Intelligence, 10, 659–671.

    Google Scholar 

  • Künsch, H. R. (1987). Intrinsic autoregressions and related models on the two-dimensional lattice, Biometrika, 74, 517–524.

    Google Scholar 

  • Mollié, A. (1990). Représentation géographique des taux de mortalité: modélisation spatiale et méthodes Bayésiennes (unpublished Ph. D. thesis).

  • Owen, A. (1989). Image segmentation via iterated conditional expectations, Tech. Report, Department of Statistics, Stanford University, California.

    Google Scholar 

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An earlier version of this article was presented at the symposium on the Analysis of Statistical Information held in the Institute of Statistical Mathematics, Tokyo during December 5–8, 1989.

This research was carried out partly at the University of Durham, U.K., with the support of an award by the Complex Stochastic Systems Initiative of the Science and Engineering Research Council.

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Besag, J., York, J. & Mollié, A. Bayesian image restoration, with two applications in spatial statistics. Ann Inst Stat Math 43, 1–20 (1991). https://doi.org/10.1007/BF00116466

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  • DOI: https://doi.org/10.1007/BF00116466

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