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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Signal recovery via TV-type energies


Authors: M. Fuchs, J. Müller and C. Tietz
Original publication: Algebra i Analiz, tom 29 (2017), nomer 4.
Journal: St. Petersburg Math. J. 29 (2018), 657-681
MSC (2010): Primary 26A45; Secondary 49J05, 49J45, 34B15
DOI: https://doi.org/10.1090/spmj/1511
Published electronically: June 1, 2018
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Abstract: 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.


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Additional Information

M. Fuchs
Affiliation: Department of Mathematics, Saarland University, P.O. Box 151150, 66041 Saarbrücken, Germany
Email: fuchs@math.uni-sb.de

J. Müller
Affiliation: Department of Mathematics, Saarland University, P.O. Box 151150, 66041 Saarbrücken, Germany
Email: jmueller@math.uni-sb.de

C. Tietz
Affiliation: Department of Mathematics, Saarland University, P.O. Box 151150, 66041 Saarbrücken, Germany
Email: tietz@math.uni-sb.de

DOI: https://doi.org/10.1090/spmj/1511
Keywords: Total variation, signal denoising, variational problems in one independent variable, linear growth, existence and regularity of solutions
Received by editor(s): February 10, 2017
Published electronically: June 1, 2018
Additional Notes: The authors thank Michael Bildhauer for many stimulating discussions
Dedicated: Dedicated to the memory of Stefan Hildebrandt
Article copyright: © Copyright 2018 American Mathematical Society

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