Martin Burger, Alexander Sawatzky, Gabriele Steidl

• Variational methods in imaging are nowadays developing towards a quite universal and exible tool, allowing for highly successful approaches on tasks like denoising, deblurring, inpainting, segmentation, super-resolution, disparity, and optical flow estimation. The overall structure of such approaches is of the form D(Ku) + alpha R(u) to min_u ; where the functional D is a data fidelity term also depending on some input data f and measuring the deviation of Ku from such and R is a regularization functional. Moreover K is a (often linear) forward operator modeling the dependence of data on an underlying image, and alpha is a positive regularization parameter. While D is often smooth and (strictly) convex, the current practice almost exclusively uses nonsmooth regularization functionals. The majority of successful techniques is using nonsmooth and convex functionals like the total variation and generalizations thereof, cf. [28, 31, 40], or l_1-norms of coeefficients arising from scalar products with some frame system, cf. [73] and references therein. The efficient solution of such variational problems in imaging demands for appropriate algorithms. Taking into account the specific structure as a sum of two very different terms to be minimized, splitting algorithms are a quite canonical choice. Consequently this field has revived the interest in techniques like operator splittings or augmented Lagrangians. In this chapter we shall provide an overview of methods currently developed and recent results as well as some computational studies providing a comparison of different methods and also illustrating their success in applications. We start with a very general viewpoint in the first sections, discussing basic notations, properties of proximal maps, firmly non-expansive and averaging operators, which form the basis of further convergence arguments. Then we proceed to a discussion of several state-of-the art algorithms and their (theoretical) convergence properties. After a section discussing issues related to the use of analogous iterative schemes for ill-posed problems, we present some practical convergence studies in numerical examples related to PET and spectral CT reconstruction.