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A Frame Decomposition of the Funk-Radon Transform

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Scale Space and Variational Methods in Computer Vision (SSVM 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14009))

Abstract

The Funk-Radon transform assigns to a function defined on the unit sphere its integrals along all great circles of the sphere. In this paper, we consider a frame decomposition of the Funk-Radon transform, which is a flexible alternative to the singular value decomposition. In particular, we construct a novel frame decomposition based on trigonometric polynomials and show its application for the inversion of the Funk-Radon transform. Our theoretical findings are verified by numerical experiments, which also incorporate a regularization scheme.

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Acknowledgements

This work was funded by the Austrian Science Fund (FWF): project F6805-N36 (SH) and the German Research Foundation (DFG): project 495365311 (MQ) within the SFB F68: “Tomography Across the Scales”. LW is partially supported by the State of Upper Austria.

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Authors and Affiliations

  1. Technische Universität Berlin, Institute of Mathematics, MA 4-3, Straße des 17. Juni 136, 10623, Berlin, Germany

    Michael Quellmalz & Paul D. Erchinger

  2. Johann Radon Institute Linz, Altenbergerstraße 69, 4040, Linz, Austria

    Lukas Weissinger & Simon Hubmer

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  1. Michael Quellmalz
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  2. Lukas Weissinger
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  3. Simon Hubmer
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  4. Paul D. Erchinger
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Correspondence to Michael Quellmalz .

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Editors and Affiliations

  1. CNRS, Université Côte d'Azur, Sophia-Antipolis, France

    Luca Calatroni

  2. University of Insubria, Como, Italy

    Marco Donatelli

  3. University of Bologna, Bologna, Italy

    Serena Morigi

  4. University of Modena and Reggio Emilia, Modena, Italy

    Marco Prato

  5. University of Genova, Genova, Italy

    Matteo Santacesaria

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Quellmalz, M., Weissinger, L., Hubmer, S., Erchinger, P.D. (2023). A Frame Decomposition of the Funk-Radon Transform. In: Calatroni, L., Donatelli, M., Morigi, S., Prato, M., Santacesaria, M. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2023. Lecture Notes in Computer Science, vol 14009. Springer, Cham. https://doi.org/10.1007/978-3-031-31975-4_4

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  • DOI: https://doi.org/10.1007/978-3-031-31975-4_4

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