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On some perturbations of the total variation image inpainting method. II: Relaxation and dual variational formulation. (English. Russian original) Zbl 1321.49054
J. Math. Sci., New York 205, No. 2, 121-140 (2015); translation from Probl. Mat. Anal. 77, 3-18 (2014).
Summary: We continue the analysis of some strongly elliptic modifications of the total variation image inpainting model formulated in the space \(\mathrm{BV}(\Omega)\) and investigate the corresponding dual variational problems. Remarkable features are the uniqueness of the dual solution and the uniqueness of the absolutely continuous part \(\nabla^a u\) of the gradient of \(\mathrm{BV}\)-solutions \(u\) on the whole domain. Additionally, any \(\mathrm{BV}\)-minimizer \(u\) automatically satisfies the inequality \(0 \leqslant u \leqslant 1\), which means that \(u\) measures the intensity of the grey level. Outside of the damaged region we even have the uniqueness of \(\mathrm{BV}\)-solutions, whereas on the damaged domain the \(L^2\)-deviation \(\| (u-v) \|_{L^2}\) of different solutions is governed by the the total variation of the singular part \(\nabla^s (u-v)\) of the vector measure \(\nabla(u-v)\). Moreover, the dual solution is related to the \(\mathrm{BV}\)-solutions through an equation of stress-strain type.
Editorial remark: for part I, see [Zbl 1321.49060].

MSC:
49N15 Duality theory (optimization)
49J45 Methods involving semicontinuity and convergence; relaxation
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
68U10 Computing methodologies for image processing
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