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Sparse Regularization of Inverse Problems by Operator-Adapted Frame Thresholding

  • We analyze sparse frame based regularization of inverse problems by means of a diagonal frame decomposition (DFD) for the forward operator, which generalizes the SVD. The DFD allows to define a non-iterative (direct) operator-adapted frame thresholding approach which we show to provide a convergent regularization method with linear convergence rates. These results will be compared to the well-known analysis and synthesis variants of sparse ℓ1-regularization which are usually implemented thorough iterative schemes. If the frame is a basis (non-redundant case), the three versions of sparse regularization, namely synthesis and analysis variants of ℓ1-regularization as well as the DFD thresholding are equivalent. However, in the redundant case, those three approaches are pairwise different.

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Metadaten
Author:Jürgen FrikelORCiD, Markus HaltmeierORCiD
DOI:https://doi.org/10.1007/978-3-030-47174-3_10
ISBN:978-3-030-47173-6
ISBN:978-3-030-47174-3
Parent Title (English):Mathematics of Wave Phenomena
Publisher:Birkhäuser; Springer International Publishing
Place of publication:Cham
Editor:Willy Dörfler, Marlis Hochbruck, Dirk Hundertmark, Wolfgang Reichel, Andreas Rieder
Document Type:Part of a Book
Language:English
Year of first Publication:2020
Release Date:2022/01/27
Edition:1st ed.
First Page:163
Last Page:178
Institutes:Fakultät Informatik und Mathematik
Begutachtungsstatus:peer-reviewed
research focus:Digitalisierung
Licence (German):Keine Lizenz - Es gilt das deutsche Urheberrecht: § 53 UrhG