Robust priors for smoothing and image restoration.

*(English)*Zbl 0805.62033Summary: The Bayesian method for restoring an image corrupted by added Gaussian noise uses a Gibbs prior for the unknown clean image. The potential of this Gibbs prior penalizes differences between adjacent grey levels. We discuss the choice of the form and the parameters of the penalizing potential in a particular example used previously by Y. Ogata [ibid. 42, No. 3, 403-433 (1990; Zbl 0719.65098)]. In this example the clean image is piecewise constant, but the constant patches and the step sizes at edges are small compared with the noise variance.

We find that contrary to results reported in Ogata the Bayesian method performs well provided the potential increases more slowly than a quadratic one and the scale parameter of the potential is sufficiently small. Convex potentials with bounded derivatives perform not much worse than bounded potentials, but are computationally much simpler. For bounded potentials we use a variant of simulated annealing. For quadratic potentials data-driven choices of the smoothing parameter are reviewed and compared. For other potentials the smoothing parameter is determined by considering which deviations from a flat image we would like to smooth out and retain respectively.

We find that contrary to results reported in Ogata the Bayesian method performs well provided the potential increases more slowly than a quadratic one and the scale parameter of the potential is sufficiently small. Convex potentials with bounded derivatives perform not much worse than bounded potentials, but are computationally much simpler. For bounded potentials we use a variant of simulated annealing. For quadratic potentials data-driven choices of the smoothing parameter are reviewed and compared. For other potentials the smoothing parameter is determined by considering which deviations from a flat image we would like to smooth out and retain respectively.

##### MSC:

62F15 | Bayesian inference |

62P99 | Applications of statistics |

62N99 | Survival analysis and censored data |

##### Keywords:

image restoration; non-Gaussian smoothness priors; maximum a posteriori estimation; images with discontinuities; Gaussian noise; Gibbs prior; penalizing potential; simulated annealing; quadratic potentials; smoothing parameter
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\textit{H. R. Künsch}, Ann. Inst. Stat. Math. 46, No. 1, 1--19 (1994; Zbl 0805.62033)

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##### References:

[1] | Akaike, H. (1980). Likelihood and Bayes procedure,Bayesian Statistics (eds. J. M. Bernardo, M. H. De Groot, D. V. Lindley and A. F. M. Smith), University Press, Valencia, Spain. · Zbl 0471.62033 |

[2] | Bellman, R. (1960).Introduction to Matrix Analysis. McGraw-Hill, New York. · Zbl 0124.01001 |

[3] | Besag, J. (1986). On the statistical analysis of dirty pictures (with discussion),J. Roy. Statist. Soc. Ser. B,48, 192-236. · Zbl 0609.62150 |

[4] | Besag, J. (1989). Towards Bayesian image analysis,J. Appl. Statist.,16, 395-407. |

[5] | Besag, J., York, J. and Mollié, A. (1991). Bayesian image restoration, with two applications in spatial statistics (with discussion),Ann. Inst. Statist. Math.,43, 1-59. · Zbl 0760.62029 |

[6] | Geman, D. and Reynolds, G. (1992). Constrained restoration and the recovery of discontinuities,IEEE Transactions on Pattern Analysis and Machine Intelligence,14, 367-383. · Zbl 05110945 |

[7] | Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,IEEE Transactions on Pattern Analysis and Machine Intelligence,6, 721-741. · Zbl 0573.62030 |

[8] | Geman, S. and McClure, D. E. (1987). Statistical methods for tomographic image reconstruction,Bulletin Internat. Statist. Inst. (Proc. 46 Session),52, Book 4, 5-21. |

[9] | Green, P. J. (1990). Penalized likelihood reconstructions from emission tomography data using a modified EM algorithm,IEEE Transactions on Medical Imaging,9, 84-93. |

[10] | Hall, P. and Titterington, D. M. (1986). On some smoothing techniques used in image restoration,J. Roy. Statist. Soc. Ser. B,48, 330-343. · Zbl 0614.62124 |

[11] | Kay, J. W. (1988). On the choice of regularisation parameter in image restoration,Pattern Recognition (ed. J. Kittler), Lecture Notes in Computer Science,301, 587-596. |

[12] | Kitagawa, G. (1987). Non-Gaussian state space modeling of nonstationary time series (with discussion),J. Amer. Statist. Assoc.,82, 1032-1063. · Zbl 0644.62088 |

[13] | Leclerc, Y. G. (1989). Constructing simple stable descriptions for image partitioning,International Journal of Computer Vision,3, 73-102. |

[14] | Ogata, Y. (1990). A Monte Carlo method for an objective Bayesian procedure,Ann. Inst. Statist. Math.,42, 403-433. · Zbl 0719.65098 |

[15] | Rousseeuw, P. J. and Leroy, A. M. (1987).Robust Regression and Outlier Detection, Wiley, New York. · Zbl 0711.62030 |

[16] | Speed, T. P. (1978). Relations between models for spatial data, contingency tables and Markov fields on graphs,Supplement Advances in Applied Probability,10, 111-122. · Zbl 0388.62056 |

[17] | Wahba, G. (1985). A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem,Ann. Statist.,13, 1378-1402. · Zbl 0596.65004 |

[18] | Wahba, G. (1990).Spline Models for Observational Data, CBMS-NSF Regional Conference Series,59, SIAM, Philadelphia. · Zbl 0813.62001 |

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