On some Bayesian choices of regularization parameter in image restoration

Recently there has been considerable interest in the problem of estimating 'optimal' degrees of smoothing within the context of Bayesian solutions to ill-posed inverse problems (e.g. Gull (1989) and Thompson and co-workers (1990)). In this paper we examine the hyperparametric method of selecting the regularization parameter described in Gull. We show how Gull's results may be generalised to situations where the log-prior may be approximated by a positive semi-definite quadratic form. We also show by means of some illustrative examples that hyperparametric approach can lead to undersmoothed reconstructions of the underlying image.