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Regularization of inverse problems by filtered diagonal frame decomposition

  • Inverse problems are at the heart of many practical problems such as image reconstruction or nondestructive testing. A characteristic feature is their instability with respect to data perturbations. To stabilize the inversion process, regularization methods must be developed and applied. In this paper, we introduce the concept of filtered diagonal frame decomposition, which extends the classical filtered SVD to the case of frames. The use of frames as generalized singular systems allows a better match to a given class of potential solutions and is also beneficial for problems where the SVD is not analytically available. We show that filtered diagonal frame decompositions yield convergent regularization methods, derive convergence rates under source conditions and prove order optimality. Our analysis applies to bounded and unbounded forward operators. As a practical application of our tools, we study filtered diagonal frame decompositions for inverting the Radon transform as an unbounded operator on L2(R2).

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Metadaten
Author:Andrea Ebner, Jürgen FrikelORCiD, D. Lorenz, J. Schwab, Markus HaltmeierORCiD
DOI:https://doi.org/10.1016/j.acha.2022.08.005
Parent Title (English):Applied and Computational Harmonic Analysis
Publisher:Elsevier
Document Type:Article
Language:English
Year of first Publication:2023
Release Date:2022/11/24
Volume:62
Issue:January
First Page:66
Last Page:83
Note:
Preprint unter: https://arxiv.org/pdf/2008.06219.pdf
OPUS-Einrag unter: https://opus4.kobv.de/opus4-oth-regensburg/admin/document/edit/id/2290
Institutes:Fakultät Informatik und Mathematik
Begutachtungsstatus:peer-reviewed
research focus:Digitalisierung
Licence (German):Keine Lizenz - Es gilt das deutsche Urheberrecht: § 53 UrhG