Investigating the Influence of Box-Constraints on the Solution of a Total Variation Model via an Efficient Primal-Dual Method

Abstract

In this paper, we investigate the usefulness of adding a box-constraint to the minimization of functionals consisting of a data-fidelity term and a total variation regularization term. In particular, we show that in certain applications an additional box-constraint does not effect the solution at all, i.e., the solution is the same whether a box-constraint is used or not. On the contrary, i.e., for applications where a box-constraint may have influence on the solution, we investigate how much it effects the quality of the restoration, especially when the regularization parameter, which weights the importance of the data term and the regularizer, is chosen suitable. In particular, for such applications, we consider the case of a squared L 2 data-fidelity term. For computing a minimizer of the respective box-constrained optimization problems a primal-dual semi-smooth Newton method is presented, which guarantees superlinear convergence.

Keywords

• box-constrained total variation minimization
• semi-smooth Newton
• image reconstruction
• automated parameter selection