Signal recovery via TV-type energies.

*(English)*Zbl 1392.49003
St. Petersbg. Math. J. 29, No. 4, 657-681 (2018) and Algebra Anal. 29, No. 4, 159-195 (2017).

Summary: One-dimensional variants are considered of the classical first order total variation denoising model introduced by Rudin, Osher, and Fatemi. This study is based on previous work of the authors on various denoising and inpainting problems in image analysis, where variational methods in arbitrary dimensions were applied. More than being just a special case, the one-dimensional setting makes it possible to study regularity properties of minimizers by more subtle methods that do not have correspondences in higher dimensions. In particular, quite strong regularity results are obtained for a class of data functions that contains many of the standard examples from signal processing such as rectangle or triangle signals as a special case. The analysis of the related Euler-Lagrange equation, which turns out to be a second order two-point boundary value problem with Neumann conditions, by ODE methods completes the picture of this investigation.

##### MSC:

49J05 | Existence theories for free problems in one independent variable |

49N60 | Regularity of solutions in optimal control |

26A45 | Functions of bounded variation, generalizations |

49J45 | Methods involving semicontinuity and convergence; relaxation |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

##### Keywords:

total variation; signal denoising; variational problems in one independent variable; linear growth; existence and regularity of solutions
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\textit{M. Fuchs} et al., St. Petersbg. Math. J. 29, No. 4, 657--681 (2018; Zbl 1392.49003)

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