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A Bayesian hierarchical model for total variation regularisation is presentedin this paper. All the parameters of an inverse problem, including the"regularisation parameter", are estimated simultaneously from the data in themodel. The model is based on the characterisation of the Laplace density prioras a scale mixture of Gaussians. With different priors on the mixture variable,other total variation like regularisations e.g. a prior that is related tot-distribution, are also obtained. An approximation of the resulting posteriormean is found using a variational Bayes method. In addition, an iterativealternating sequential algorithm for computing the maximum a posterioriestimate is presented. The methods are illustrated with examples of imagedeblurring. Results show that the proposed model can be used for automaticedge-preserving inversion in the case of image deblurring. Despite promisingresults, some difficulties with the model were encountered and are subject tofuture work.

Bayesian Hierarchical Model of Total Variation Regularisation for Image Deblurring