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On some perturbations of the total variation image inpainting method. I: Regularity theory. (English. Russian original) Zbl 1321.49060
J. Math. Sci., New York 202, No. 2, 154-169 (2014); translation from Probl. Mat. Anal. 76, 39-52 (2014).
Summary: We investigate modifications of the total variation image inpainting model and discuss the existence as well as the smoothness of solutions to these new variational problems. We emphasize that our variant of a linear growth regularization is completely discussed in the Sobolev space \(W_{1}^{1}\) without passing to a relaxed version in the space of functions with bounded variation. Our analysis mainly concentrates on the theoretical background of the problems under consideration. Nevertheless, first computational experiments indicate that our model is numerically comparable with the TV-model with the advantage of a clear underlying analysis presented in the following.
Editorial remark: for part II, see [Zbl 1321.49054].

MSC:
49N60 Regularity of solutions in optimal control
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
68U10 Computing methodologies for image processing
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