Abstract
The Bayesian derivation of “Classic” MaxEnt image processing (Skilling 1989a) shows that exp(αS(f,m)), where S(f,m) is the entropy of image f relative to model m, is the only consistent prior probability distribution for positive, additive images. In this paper the derivation of “Classic” MaxEnt is completed, showing that it leads to a natural choice for the regularising parameter α, that supersedes the traditional practice of setting x2=N. The new condition is that the dimensionless measure of structure -2αS should be equal to the number of good singular values contained in the data. The performance of this new condition is discussed with reference to image deconvolution, but leads to a reconstruction that is visually disappointing. A deeper hypothesis space is proposed that overcomes these difficulties, by allowing for spatial correlations across the image.
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References
Charter, M.K. & Gull, S.F. (1988) Maximum entropy and its application to the calculation of drug absorption rates. J. Pharmacokinetics and Biopharmaceutics, 15, 645–655.
Daniell, G.J. & Gull, S.F. (1980). Maximun entropy algorithm applied to image enhancement, IEE Proc, 127E, 170–172.
Davies, A.R. & Anderssen, R.S. (1986). Optimisation in the regularisation of ill-posed problems. J. Austral. Math. Soc. Ser. B., 28, 114–133.
Frieden, B.R. (1972). Restoring with maximum likelihood and maximum entropy. J. Opt. Soc. Am., 62, 511–518.
Geman, S. & Geman, D. (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Trans PAMI-6, 721–741
Gull, S.F. & Daniell, G.J. (1978). Image reconstruction from incomplete and noisy data. Nature, 272, 686–690.
Gull, S.F. & Daniell, G.J. (1979). The maximum entropy method. In Image Formation from Coherence Functions in Astronomy, ed. C. van Schooneveld, pp. 219–225, Reidel.
Gull, S.F. & Skilling, J. (1984). Maximum entropy method in image processing. IEE Proc, 131(F), 646–659.
Home, K. (1985) Images of accretion discs I: The eclipse mapping method, Mon. Not. R. astr. Soc, 213, 129–141.
Jaynes, E.T. (1978). Where do we stand on maximum entropy? Reprinted in E.T. Jaynes: Papers on Probability, Statistics and Statistical Physics, ed. R. Rosenkrantz, pp 211–314. Dordrecht 1983: Reidel.
Jaynes, E.T. (1986a). Bayesian methods: general background. In Maximum Entropy and Bayesian Methods in Applied Statistics, ed. J.H. Justice., pp 1–25. Cambridge University Press.
Jaynes, E.T. (1986b). Monkeys, kangaroos and N. In Maximum Entropy and Bayesian Methods in Applied Statistics, ed. J.H. Justice, pp. 26–58, Cambridge Univ. Press.
Kinderman, R. & Snell, J.L. (1980) Markov random fields and their applications. Amer. Math. Soc. Providence, RI.
Marsh, T.R. & Horne, K. (1989) Maximum entropy tomography of accretion discs from their emission lines. In these Proceedings.
Skilling, J. (1989a). Classic Maximum Entropy. In these Proceedings.
Skilling, J. (1989b). The eigenvalues of mega-dimensional matrices. In these Proceedings. Tikhonov, A.N. & Arsenin, V.Y. (1977). Solutions of ill-posed problems. Wiley, New York.
Titterington, D.M. (1985) General structure of regularisation procedures in image reconstruction. Astron. Astrophys. 144, 381–387.
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© 1989 Springer Science+Business Media Dordrecht
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Gull, S.F. (1989). Developments in Maximum Entropy Data Analysis. In: Skilling, J. (eds) Maximum Entropy and Bayesian Methods. Fundamental Theories of Physics, vol 36. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7860-8_4
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DOI: https://doi.org/10.1007/978-94-015-7860-8_4
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