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The L1-Potts functional for robust jump-sparse reconstruction.
SIAM J. Numer. Anal. 53, 644-673 (2015)
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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Publication type Article: Journal article
Document type Scientific Article
Keywords Potts Functional ; Jump-sparse Reconstruction ; Nonconvex Functional ; Penalized Absolute Deviation ; Adaptive Estimation; Least-squares Estimators; Images
ISSN (print) / ISBN 0036-1429
Quellenangaben Volume: 53, Issue: 1, Pages: 644-673
Publisher Society for Industrial and Applied Mathematics (SIAM)
Publishing Place Philadelphia
Reviewing status Peer reviewed
Institute(s) Institute of Computational Biology (ICB)