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 variation minimization problem. In particular, the predual total variation minimization problem is decomposed into overlapping domains yielding subdomain problems in the respective dual space. Subsequently the

The minimization of a functional consisting of a combined L1/L2 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 experim

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., f

Choices of regularization parameters are central to variational methods for image restoration. In this paper, a spatially adaptive (or distributed) regularization scheme is developed based on localized residuals, which properly balances the regularization weight between regions containing image details and homogeneous regions. Surrogate iterative methods are employed to handle given subsampled dat

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 m

Based on the weighted total variation model and its analysis pursued in Hintermüller and Rautenberg 2016, in this paper a continuous, i.e., infinite dimensional, projected gradient algorithm and its convergence analysis are presented. The method computes a stationary point of a regularized bilevel optimization problem for simultaneously recovering the image as well as determining a spatially distr

Algorithms for automatically selecting a scalar or locally varying regularization parameter for total variation models with an Lτ-data fidelity term, τ∈ { 1 , 2 } , are presented. The automated selection of the regularization parameter is based on the discrepancy principle, whereby in each iteration a total variation model has to be minimized. In the case of a locally varying parameter, this amoun

In this paper non-overlapping domain decomposition methods for the pre-dual total variation minimization problem are introduced. Both parallel and sequential approaches are proposed for these methods for which convergence to a minimizer of the original problem is established. The associated subproblems are solved by a semi-smooth Newton method. Several numerical experiments are presented, which sh

The minimization of a functional composed of a nonsmooth and nonadditive regularization term and a combined L1 and L2 data-fidelity term is proposed. It is shown analytically and numerically that the new model has noticeable advantages over popular models in image processing tasks. For the numerical minimization of the new objective, subspace correction methods are introduced which guarantee the c

In this paper a multimodal optical-resolution photoacoustic and fluorescence microscope in frequency domain is presented. Photoacoustic waves and modulated fluorescence are generated in chromophores by using a modulated diode laser. The photoacoustic waves, recorded with a hydrophone, and the fluorescence signals, acquired with an avalanche photodiode, are simultaneously measured using a lock-in t

In this paper, we show additional properties of the limit of a sequence produced by the subspace correction algorithm proposed by Fornasier and Schönlieb [SIAM J. Numer. Anal., 47 (2009), pp. 3397-3428 for L2/TV-minimization problems. An important but missing property of such a limiting sequence in that paper is the convergence to a minimizer of the original minimization problem, which was obtaine

Computational problems of large-scale data are gaining attention recently due to better hardware and hence, higher dimensionality of images and data sets acquired in applications. In the last couple of years non-smooth minimization problems such as total variation minimization became increasingly important for the solution of these tasks. While being favorable due to the improved enhancement of im

In this paper we are concerned with the analysis of convergent sequential and parallel overlapping domain decomposition methods for the minimization of functionals formed by a discrepancy term with respect to the data and a total variation constraint. To our knowledge, this is the first successful attempt of addressing such a strategy for the nonlinear, nonadditive, and nonsmooth problem of total