Fourier Analysis of Periodic Radon Transforms
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(2020) 26:64
Fourier Analysis of Periodic Radon Transforms Jesse Railo1 Received: 9 September 2019 / Revised: 29 June 2020 © The Author(s) 2020
Abstract We study reconstruction of an unknown function from its d-plane Radon transform on the flat torus Tn = Rn /Zn when 1 ≤ d ≤ n − 1. We prove new reconstruction formulas and stability results with respect to weighted Bessel potential norms. We solve the associated Tikhonov minimization problem on H s Sobolev spaces using the properties of the adjoint and normal operators. One of the inversion formulas implies that a compactly supported distribution on the plane with zero average is a weighted sum of its X-ray data. Keywords Radon transform · Fourier analysis · Periodic distributions · Regularization Mathematics Subject Classification 44A12 · 42B05 · 46F12 · 45Q05
1 Introduction We study reconstruction of an unknown function from its d-plane Radon transform on the flat torus Tn = Rn /Zn when 1 ≤ d ≤ n − 1. The d-plane Radon transform of a function f on Tn encodes the integrals of f over all periodic d-planes. The usual d-plane Radon transform of compactly supported objects on Rn can be reduced into the periodic d-plane Radon transform, but not vice versa. This was demonstrated for the geodesic X-ray transform in the recent work of Ilmavirta et al. [11]. As general references on the Radon transforms, we point to [5,6,14,15]. Reconstruction formulas for integrable functions and a family of regularization strategies considered in this article were derived in [11] for the geodesic X-ray transform (d = 1) on T2 . We extend these methods to the d-plane Radon transforms of
Communicated by Eric Todd Quinto.
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Jesse Railo [email protected] Department of Mathematics and Statistics, University of Jyväskylä, P.O.Box 35 (MaD), 40014 Jyväskylä, Finland 0123456789().: V,-vol
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Journal of Fourier Analysis and Applications
(2020) 26:64
higher dimensions, study new types of reconstruction formulas for distributions, and prove new stability estimates on the Bessel potential spaces. This article considers only the mathematical theory of Radon transforms on Tn , whereas numerical algorithms (Torus CT) were implemented in [11,13]. Injectivity, a reconstruction method and certain stability estimates of the d-plane Radon transform on Tn were proved for distributions by Ilmavirta in [7]. Our reconstruction formulas and stability estimates in this article are different than the ones in [7]. The first injectivity result for the geodesic X-ray transform on T2 was obtained by Strichartz in [19], and generalized to Tn by Abouelaz and Rouvière in [2] if the Fourier transform is 1 (Zn ). Abouelaz proved uniqueness under the same assumption for the d-plane Radon transform in [1]. The X-ray transform and tensor tomography on Tn has been applied to other integral geometry problems. These examples include the broken ray transform on boxes [7], the geodesic ray transform on Lie groups [8], tensor tomography on periodic slabs [10], and the ray transforms on Minkowski tori [9
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