Generalized Abel equations and applications to translation invariant Radon transforms

Generalized Abel equations have been employed in the recent literature to invert Radon transforms which arise in a number of important imaging applications, including Compton Scatter Tomography (CST), Ultrasound Reflection Tomography (URT), and X-ray CT. In this paper, we present novel injectivity r...

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Bibliographic Details
Published inJournal of inverse and ill-posed problems Vol. 32; no. 4; pp. 835 - 857
Main Author Webber, James W.
Format Journal Article
LanguageEnglish
Published De Gruyter 01.08.2024
Subjects
45Dxx
Abel equations
Radon transforms
uniqueness of solution
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Summary:Generalized Abel equations have been employed in the recent literature to invert Radon transforms which arise in a number of important imaging applications, including Compton Scatter Tomography (CST), Ultrasound Reflection Tomography (URT), and X-ray CT. In this paper, we present novel injectivity results and inversion methods for generalized Abel operators. We apply our theory to a new Radon transform, , of interest in URT, which integrates a square integrable function of compact support, 𝑓, over ellipsoid and hyperboloid surfaces with centers on a plane. Using our newly established theory on generalized Abel equations, we show that is injective and provide an inversion method based on Neumann series. In addition, using algebraic methods, we present image phantom reconstructions from data with added pseudo-random noise.
ISSN:0928-0219
1569-3945
DOI:10.1515/jiip-2023-0049