Bayesian Hierarchical Model of Total Variation Regularisation for Image Deblurring

TL;DR

This paper introduces a Bayesian hierarchical model for total variation regularisation in image deblurring, estimating all parameters from data and demonstrating automatic edge-preserving inversion with promising results.

Contribution

It presents a novel Bayesian hierarchical framework that estimates regularisation parameters automatically and extends total variation regularisation using scale mixtures of Gaussians.

Findings

Effective automatic edge-preserving deblurring demonstrated

Variational Bayes used for posterior mean approximation

Model shows promising results with some limitations

Abstract

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.