Variational methods are amongst the most powerful tools for tackling ill-posed problems in the field of mathematical image processing. They contribute significantly to modern imaging applications such as magnetic resonance imaging (MRI) or computed tomography (CT). Regularisation is the mathematical key concept responsible for the success of these methods.
After a brief introduction to variational methods and applications in image processing, this talk will consider the analysis of measure based regularizers, in particular the Total Generalized Variation (TGV) functional. The main idea behind this functional is to enforce piecewise smoothness while still allowing jump discontinuities by employing a weak form of higher order differentiation. A particular focus will be put on the context of linear inverse problems with the application of image reconstruction from under-sampled MR data. Possible extensions to the important yet less investigated question of appropriate regularization for image sequence will also be mentioned.