Energy-based image simplification with nonlocal data and smoothness terms

Image simplification and smoothing is a very important basic ingredient of a lot of practical applications. In this paper we compare different numerical approaches to solve this image approximation task within a unifying variational approach presented in [8]. For methods based on fixed point iterations we show the existence of fixed points. To speed up the convergence we also use two approaches involving Newton's method which is only applicable for convex penalisers. The running time in practice is studied with numerical examples in 1-D and 2-D.