Andreas Langer

A convergent overlapping domain decomposition method for total variation minimization

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 variation minimization. We provide several numerical experiments, showing the successful application of the algorithm for the restoration of 1D signals and 2D images in interpolation/inpainting problems, respectively, and in a compressed sensing problem, for recovering piecewise constant medical-type images from partial Fourier ensembles.

Subspace Correction Methods for a Class of Nonsmooth and Nonadditive Convex Variational Problems with Mixed

The minimization of a functional composed of a nonsmooth and nonadditive regularization term and a combined

L^1/L^2

L^1

Data-Fidelity in Image Processing

and

Bregmanized Domain Decomposition for Image Restoration

L^2

Non-Overlapping Domain Decomposition Methods For Dual Total Variation Based Image Denoising

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 convergence and monotone decay of the associated energy along the iterates. Moreover, an estimate of the distance between the outcome of the subspace correction method and the global minimizer of the nonsmooth objective is derived. This estimate and numerical experiments for image denoising, inpainting, and deblurring indicate that in practice the proposed subspace correction methods indeed approach the global solution of the underlying minimization problem.

Wavelet Decomposition Method for L2/TV-Image Deblurring ∗

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 images compared to smooth imaging approaches, non-smooth minimization problems typically scale badly with the dimension of the data. Hence, for large imaging problems solved by total variation minimization domain decomposition algorithms have been proposed, aiming to split one large problem into N>1 smaller problems which can be solved on parallel CPUs. The N subproblems constitute constrained minimization problems, where the constraint enforces the support of the minimizer to be the respective subdomain.In this paper we discuss a fast computational algorithm to solve domain decomposition for total variation minimization. In particular, we accelerate the computation of the subproblems by nested Bregman iterations. We propose a Bregmanized Operator Splitting–Split Bregman (BOS-SB) algorithm, which enforces the restriction onto the respective subdomain by a Bregman iteration that is subsequently solved by a Split Bregman strategy. The computational performance of this new approach is discussed for its application to image inpainting and image deblurring. It turns out that the proposed new solution technique is up to three times faster than the iterative algorithm currently used in domain decomposition methods for total variation minimization.

Automated Parameter Selection for Total Variation Minimization in Image Restoration

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 show the successful application of the sequential and parallel algorithm for image denoising.

Optimal Selection of the Regularization Function in a Weighted Total Variation Model. Part II: Algorithm, Its Analysis and Numerical Tests

In this paper, we show additional properties of the limit of a sequence produced by the subspace correction algorithm proposed by Fornasier and Schonlieb [SIAM J. Numer. Anal., 47 (2009), pp. 3397--3428] for

Frequency domain optical resolution photoacoustic and fluorescence microscopy using a modulated laser diode

L_2/