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Existence of generalized minimizers and dual solutions for a class of variational problems with linear growth related to image recovery. (English. Russian original) Zbl 1331.49014
J. Math. Sci., New York 210, No. 4, 458-475 (2015); translation from Probl. Mat. Anal. 81, 107-120 (2015).
Summary: We continue the analysis of modifications of the total variation image inpainting method formulated on the space \(BV(\Omega)^M\) 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 \(\Omega\) 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 \(\nabla^au\) of the gradient of \(BV\)-solutions \(u\) on the domain \(\Omega\) 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.

49J45 Methods involving semicontinuity and convergence; relaxation
49N15 Duality theory (optimization)
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
Full Text: DOI
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