On the solvability in Sobolev spaces and related regularity results for a variant of the TV-image recovery model: the vector-valued case.

*(English)*Zbl 1382.49044Summary: We study classes of variational problems with energy densities of linear growth acting on vector-valued functions. Our energies are strictly convex variants of the TV-regularization model introduced by L. I. Rudin et al. [Physica D 60, No. 1–4, 259–268 (1992; Zbl 0780.49028)] as a powerful tool in the field of image recovery. In contrast to our previous work we here try to figure out conditions under which we can solve these variational problems in classical spaces, e.g. in the Sobolev class \(W^{1,1}\).

##### MSC:

49N60 | Regularity of solutions in optimal control |

62H35 | Image analysis in multivariate analysis |

49K27 | Optimality conditions for problems in abstract spaces |

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

##### Keywords:

variational problems of linear growth; TV-regularization; denoising and inpainting of multicolor images; existence of solutions in Sobolev spaces
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\textit{M. Bildhauer} et al., J. Elliptic Parabol. Equ. 2, No. 1--2, 341--355 (2016; Zbl 1382.49044)

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##### References:

[1] | R. A. Adams. Sobolev spaces, volume 65 of Pure and Applied Mathematics. Academic Press, New-York-London, 1975. |

[2] | L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discontinuity problems. Clarendon Press, Oxford, 2000. · Zbl 0957.49001 |

[3] | Anzelotti, G; Giaquinta, M, Convex functionals and partial regularity, Archive for Rational Mechanics and Analysis, 102, 243-272, (1988) · Zbl 0658.49005 |

[4] | M. Bildhauer. Convex Variational Problems: Linear, nearly Linear and Anisotropic Growth Conditions. Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2003. · Zbl 1033.49001 |

[5] | Bildhauer, M; Fuchs, M, A variational approach to the denoising of images based on different variants of the TV-regularization, Appl. Math. Optim., 66, 331-361, (2012) · Zbl 1260.49074 |

[6] | Bildhauer, M; Fuchs, M, On some perturbations of the total variation image inpainting method. part I: regularity theory, J. Math. Sciences, 202, 154-169, (2014) · Zbl 1321.49060 |

[7] | Bildhauer, M; Fuchs, M, On some perturbations of the total variation image inpainting method. part II: relaxation and dual variational formulation, J. Math. Sciences, 2052, 121-140, (2015) · Zbl 1321.49054 |

[8] | Bildhauer, M; Fuchs, M; Tietz, C, C1,a-interior regularity for minimizers of a class of variational problems with linear growth related to image inpainting, Algebra i Analiz, 27, 51-65, (2015) · Zbl 1335.49058 |

[9] | M. M. Fuchs, and J. Weickert. Denoising and inpainting of images using TV-type energies. to appear in J. Math. Sciences. · Zbl 1381.94011 |

[10] | Fuchs, M; Müller, J, A higher order TV-type variational problem related to the denoising and inpainting of images, (2016) · Zbl 1358.49043 |

[11] | M. J. Müller, and C. Tietz. Signal recovery via TV-type energies, 2016. Technical Report No. 381, Department of Mathematics, Saarland University. · Zbl 1392.49003 |

[12] | Fuchs, M; Tietz, C, Existence of generalized minimizers and of dual solutions for a class of variational problems with linear growth related to image recovery, J. Math. Sciences, 210, 458-475, (2015) · Zbl 1331.49014 |

[13] | E. Giusti. Minimal surfaces and functions of bounded variation, volume 80 of Monographs in Mathematics. Birkhäuser, Basel, 1984. · Zbl 0545.49018 |

[14] | J. Müller and C. Tietz. Existence and almost everywhere regularity of generalized minimizers for a class of variational problems with linear growth related to image inpainting, 2015. Technical Report No. 363, Department of Mathematics, Saarland University. |

[15] | Rudin, L I; Osher, S; Fatemi, E, Nonlinear total variation based noise removal algorithms, Physica D, 60, 259-268, (1992) · Zbl 0780.49028 |

[16] | Schmidt, T, Partial regularity for degenerate variational problems and image restoration models in BV, Indiana Univ. Math. J., 63, 213-279, (2014) · Zbl 1319.49058 |

[17] | C. Tietz. Existence and regularity theorems for variants of the TV-image inpainting method in higher dimensions and with vector-valued data. PhD thesis, Saarland University, 2016. |

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