The minimization of a functional consisting of a combined L ¹ / L ² -data-fidelity term and a total variation term, named L ¹ - L ² -TV model, is considered to remove a mixture of Gaussian and impulse noise in images, which are possibly additionally deformed by some convolution operator. We investigate analytically the stability of this model with respect to its parameters and link it to a constrained minimization problem. Based on these investigations and a statistical characterization of the mixed Gaussian-impulse noise a fully automated parameter selection algorithm for the L ¹ - L ² -TV model is presented. It is shown by numerical experiments that the proposed method finds parameters with which noise is removed considerably while features are preserved in images.

Alternating and parallel overlapping domain decomposition methods for the minimization of the total variation are presented. Their derivation is based on the predual formulation of the total variat...

The minimization of a functional consisting of a combined $L^1$/ $L^2$ data fidelity term and a total variation regularization term with a locally varying regularization parameter for the removal of mixed Gaussian–impulse noise is considered. Based on a related locally constrained optimization problem, algorithms for automatically selecting the spatially varying parameter are presented. Numerical experiments for image denoising are shown, which demonstrate that the locally varying parameter selection algorithms are able to generate solutions which are of higher restoration quality than solutions obtained with scalar parameters.

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.