  This paper presents a sparse Bayesian learning algorithm for inverse problems
in signal and image processing with a total variation (TV) sparsity prior.
Because of the prior used, and the fact that the prior parameters are estimated
directly from the data, sparse Bayesian learning often produces more accurate
results than the typical maximum a posteriori Bayesian estimates for sparse
signal recovery. It also provides a full posterior distribution. However,
sparse Bayesian learning is only available to problems with a direct sparsity
prior or those formed via synthesis. This paper demonstrates how a problem with
a TV sparsity prior can be formulated in a synthesis approach. We then develop
a method that combines this synthesis-based TV with the sparse Bayesian
learning algorithm and provide numerical examples to demonstrate how our new
technique is effectively employed.
