We continue the analysis of modifications of the total variation image inpainting method formulated on the space BV (Ω)M and treat the case of vector-valued images where we do not impose any structure condition on the density F and the dimension of the domain Ω is arbitrary. We discuss the existence of generalized solutions of the corresponding variational problem and show the unique solvability of the associated dual variational problem. We establish the uniqueness of the absolutely continuous part ∇ a u of the gradient of BV -solutions u on the domain Ω and get the uniqueness of BV -solutions outside the damaged region D. We also prove new density results for functions of bounded variation and for Sobolev functions. Bibliography: 36 titles.
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Translated from Problemy Matematicheskogo Analiza 81, August 2015, pp. 107–120.
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Fuchs, M., Tietz, C. Existence of Generalized Minimizers and Dual Solutions for a Class of Variational Problems with Linear Growth Related to Image Recovery. J Math Sci 210, 458–475 (2015). https://doi.org/10.1007/s10958-015-2575-2
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DOI: https://doi.org/10.1007/s10958-015-2575-2