Existence and regularity theorems for variants of the TV-image inpainting method in higher dimensions with vector-valued data

摘要

In this thesis we are mainly concerned with a modification of the classical total variation image inpainting model. This alteration, which leads to a variational problem with linear growth, has been suggested by M. Bildhauer and M. Fuchs and is of interest since it describes inpainting with simultaneous denoising, i.e., we jointly reconstruct the region of the image for which data are missing or inaccessible and denoise the generated image on the entire domain. First numerical experiments in collaboration with J. Weickert have revealed that the above modification is numerically comparable to the standard total variation image inpainting model with the advantage of a comprehensive existence and regularity theory of the corresponding solutions. The main focus of this thesis lies on establishing such a theory for any dimension together with arbitrary codimension, i.e., vector-valued images are included in our investigations.More precisely we first show existence of generalized minimizers (in a suitable sense) and pass to the associated dual problem. In this context we prove new density results for functions of bounded variation and for Sobolev functions. Afterwards we investigate the regularity behavior of generalized minimizers.As a slight advancement we moreover study a special non-autonomous variant of the above variational problem in the context of the denoising of images for which we establish existence and regularity results of generalized minimizers.