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A higher order TV-type variational problem related to the denoising and inpainting of images. (English) Zbl 1358.49043
Summary: We give a comprehensive survey on a class of higher order variational problems which are motivated by applications in mathematical imaging. The overall aim of this note is to investigate if and in which manner results from the first author’s previous work on variants of the TV-regularization model (see e.g. M. Bildhauer and M. Fuchs [Appl. Math. Optim. 66, No. 3, 331–361 (2012; Zbl 1260.49074); J. Math. Sci., New York 202, No. 2, 154–169 (2014; Zbl 1321.49060); translation from Probl. Mat. Anal. 76, 39–52 (2014); ibid. 205, No. 2, 121–140 (2015; Zbl 1321.49054); translation from Probl. Mat. Anal. 77, 3–18 (2014)] and M. Fuchs and C. Tietz [ibid. 210, No. 4, 458–475 (2015; Zbl 1331.49014); translation from Probl. Mat. Anal. 81, 107–120 (2015)]) can be extended to functionals which involve higher derivatives. This seems to be not only of theoretical interest, but also relevant to applications since higher order TV-denoising appears to maintain the advantages of the classical model as introduced by L. I. Rudin et al. [Physica D 60, No. 1–4, 259–268 (1992; Zbl 0780.49028)] while avoiding the unpleasant “staircasing” effect (see e.g. [K. Bredies et al., SIAM J. Imaging Sci. 3, No. 3, 492–526 (2010; Zbl 1195.49025)] or [M. Lysaker et al., IEEE Trans. Image Process. 12, No. 12, 1579–1590 (2003; Zbl 1286.94020)]). Our paper features results concerning generalized solutions in spaces of functions of higher order bounded variation, dual solutions as well as partial regularity of minimizers.

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
49N60 Regularity of solutions in optimal control
49N15 Duality theory (optimization)
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
62H35 Image analysis in multivariate analysis
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