  A Bayesian hierarchical model for total variation regularisation is presented
in this paper. All the parameters of an inverse problem, including the
"regularisation parameter", are estimated simultaneously from the data in the
model. The model is based on the characterisation of the Laplace density prior
as a scale mixture of Gaussians. With different priors on the mixture variable,
other total variation like regularisations e.g. a prior that is related to
t-distribution, are also obtained. An approximation of the resulting posterior
mean is found using a variational Bayes method. In addition, an iterative
alternating sequential algorithm for computing the maximum a posteriori
estimate is presented. The methods are illustrated with examples of image
deblurring. Results show that the proposed model can be used for automatic
edge-preserving inversion in the case of image deblurring. Despite promising
results, some difficulties with the model were encountered and are subject to
future work.
