Tietz, Christian
[Author]
;
Fuchs, Martin
[Contributor]
Existence and regularity theorems for variants of the TV-image inpainting method in higher dimensions with vector-valued data ; Existenz-und Regularitätssätze für Modifikationen des TV-image inpainting Modells in höheren Dimensionen mit vektorwertigen Daten
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Media type:
Doctoral Thesis;
Electronic Thesis;
E-Book
Title:
Existence and regularity theorems for variants of the TV-image inpainting method in higher dimensions with vector-valued data ; Existenz-und Regularitätssätze für Modifikationen des TV-image inpainting Modells in höheren Dimensionen mit vektorwertigen Daten
Contributor:
Tietz, Christian
[Author]
Published:
Scientific publications of the Saarland University (UdS), 2016
Footnote:
Diese Datenquelle enthält auch Bestandsnachweise, die nicht zu einem Volltext führen.
Description:
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. ; Diese Arbeit beschäftigt sich hauptsächlich mit einer Abwandlung des klassischen TV-image inpainting Modells. Diese Modifikation, welche ein Variationsproblem mit linearem Wachstum beschreibt, wurde von M. Bildhauer und M. Fuchs vorgeschlagen und vereinigt das sogenannte inpainting mit simultanem Entrauschen. Numerische Experimente in Zusammenarbeit mit J. Weickert haben gezeigt, dass die obige Modifikation im Vergleich zu den bekannten TV-image inpainting Verfahren numerisch vergleichbare Ergebnisse erzielt. Ein klarer Vorteil des neuen Modells ist jedoch, ...