Parameter estimation for hybrid wavelet-total variation regularization

In many image restoration/reconstruction problems, using redundant linear decompositions also named as frames may be fruitful. Moreover, Total Variation (TV) is also widely used in the edge-preserving regularization literature. Associating these two tools in a joint regularization framework may be of great interest since they are somehow complementary. However, estimating the regularization parameters in this case becomes a tricky issue which cannot be performed by using standard estimators. In this work, a hierarchical model is introduced to solve this problem within a fully Bayesian framework. A hybrid MCMC algorithm is subsequently proposed to sample from the derived posterior distribution. We show that this algorithm allows the regularization parameters to be determined accurately. We finally investigate its application to parallel MRI reconstruction, where the use of a joint wavelet-TV regularization is also novel.