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Weinmann, A. ; Storath, M. ; Demaret, L.

The L1-Potts functional for robust jump-sparse reconstruction.

SIAM J. Numer. Anal. 53, 644-673 (2015)
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We investigate the nonsmooth and nonconvex $L^1$-Potts functional in discrete and continuous time. We show $\Gamma$-convergence of discrete $L^1$-Potts functionals toward their continuous counterpart and obtain a convergence statement for the corresponding minimizers as the discretization gets finer. For the discrete $L^1$-Potts problem, we introduce an $O(n^2)$ time and $O(n)$ space algorithm to compute an exact minimizer. We apply $L^1$-Potts minimization to the problem of recovering piecewise constant signals from noisy measurements $f.$ It turns out that the $L^1$-Potts functional has a quite interesting blind deconvolution property. In fact, we show that mildly blurred jump-sparse signals are reconstructed by minimizing the $L^1$-Potts functional. Furthermore, for strongly blurred signals and a known blurring operator, we derive an iterative reconstruction algorithm.
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Publikationstyp Artikel: Journalartikel
Dokumenttyp Wissenschaftlicher Artikel
Schlagwörter Potts Functional ; Jump-sparse Reconstruction ; Nonconvex Functional ; Penalized Absolute Deviation ; Adaptive Estimation; Least-squares Estimators; Images
ISSN (print) / ISBN 0036-1429
e-ISSN 1095-7170
Zeitschrift SIAM Journal on Numerical Analysis
Quellenangaben Band: 53, Heft: 1, Seiten: 644-673 Artikelnummer: , Supplement: ,
Verlag Society for Industrial and Applied Mathematics (SIAM)
Verlagsort Philadelphia
Begutachtungsstatus Peer reviewed
Institut(e) Institute of Computational Biology (ICB)
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