A higher order TV-type variational problem related to the denoising and inpainting of images

We give a comprehensive survey on a class of higher order variational problems which are motivated by applications in mathematical imaging. The overall aim of this note is to investigate if and in which manner results from the first author's previous work on variants of the TV-regularization model (see e.g. Bildhauer and Fuchs (2012, 2014, 2015) and Fuchs and Tietz (2015)) can be extended to functionals which involve higher derivatives. This seems to be not only of theoretical interest, but also relevant to applications since higher order TV-denoising appears to maintain the advantages of the classical model as introduced by Rudin, Osher and Fatemi in (Rudin et al., 1992) while avoiding the unpleasant “staircasing” effect (see e.g. Bredies et al. (2010) or Lysaker et al. (2003)). Our paper features results concerning generalized solutions in spaces of functions of higher order bounded variation, dual solutions as well as partial regularity of minimizers.