Functional-Analytic and Numerical Issues in Splitting Methods for Total Variation-based Image Reconstruction

Variable splitting schemes for the function space TV-model in its primal and pre-dual formulations are considered. In the primal splitting formulation, while existence of a solution cannot be guaranteed, it is shown that quasi-minimizers of the penalized problem are asymptotically related to the solution of the original TV-model. On the other hand, for the pre-dual formulation a family of parametrized problems is introduced and a parameter dependent contraction of an associated fixed point iteration is established. Moreover, the theory is validated by numerical tests. Additionally, the augmented Lagrangian approach is studied, details on an implementation on a staggered grid are provided and numerical tests are shown.