  Regularization approaches have demonstrated their effectiveness for solving
ill-posed problems. However, in the context of variational restoration methods,
a challenging question remains, which is how to find a good regularizer. While
total variation introduces staircase effects, wavelet domain regularization
brings other artefacts, e.g. ringing. However, a compromise can be found by
introducing a hybrid regularization including several terms non necessarily
acting in the same domain (e.g. spatial and wavelet transform domains). We
adopt a convex optimization framework where the criterion to be minimized is
split in the sum of more than two terms. For spatial domain regularization,
isotropic or anisotropic total variation definitions using various gradient
filters are considered. An accelerated version of the Parallel ProXimal
Algorithm is proposed to perform the minimization. Some difficulties in the
computation of the proximity operators involved in this algorithm are also
addressed in this paper. Numerical experiments performed in the context of
Poisson data recovery, show the good behavior of the algorithm as well as
promising results concerning the use of hybrid regularization techniques.
