Fourier analysis of periodic Radon transforms

TL;DR

This paper develops new Fourier-based reconstruction formulas and stability results for the Radon transform on the flat n-torus, enhancing understanding of function recovery from plane integrals.

Contribution

It introduces novel inversion formulas and stability analysis for the Radon transform on the n-torus, and solves the Tikhonov minimization problem in Sobolev spaces.

Findings

New reconstruction formulas for the Radon transform on the n-torus

Stability results in weighted Bessel potential norms

Solution of Tikhonov minimization in Sobolev spaces

Abstract

We study reconstruction of an unknown function from its $d$-plane Radon transform on the flat $n$-torus when $1 \leq d \leq n-1$. 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.