Torus computed tomography

Abstract

We present a new computed tomography (CT) method for inverting the Radon transform in 2D. The idea relies on the geometry of the flat torus, hence we call the new method Torus CT. We prove new inversion formulas for integrable functions, solve a minimization problem associated to Tikhonov regularization in Sobolev spaces and prove that the solution operator provides an admissible regularization strategy with a quantitative stability estimate. This regularization is a simple post-processing low-pass filter for the Fourier series of a phantom. We also study the adjoint and the normal operator of the X-ray transform on the flat torus. The X-ray transform is unitary on the flat torus. We have implemented the Torus CT method using Matlab and tested it with simulated data with promising results. The inversion method is meshless in the sense that it gives out a closed form function that can be evaluated at any point of interest.

Fourier analysis of periodic Radon transforms

“…The usual d-plane Radon transform of compactly supported objects on R n can be reduced into the periodic d-plane Radon transform, but not vice versa. This was demonstrated for the geodesic Xray transform in the recent work of Ilmavirta, Koskela and Railo [10]. As general references on the Radon transforms, we point to [5,15,6,14].…”

“…Reconstruction formulas for integrable functions and a family of regularization strategies considered in this article were derived in [10] for the geodesic X-ray transform (d = 1) on T 2 . We extend these methods to the d-plane Radon transforms of higher dimensions, study new types of reconstruction formulas for distributions, and prove new stability estimates on the Bessel potential spaces.…”

“…We extend these methods to the d-plane Radon transforms of 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 T n , whereas numerical algorithms (Torus CT) were implemented in [10,13].…”

“…Here we only briefly introduce the used notation, and more details are given in subsequent sections. One can also find more details in [7,10]. Let n, d ∈ Z be such that n ≥ 2 and 1 ≤ d ≤ n − 1.…”

“…Remark 1.1. If d = n − 1, then weights are not that important for the analysis of R d as demonstrated in the case of n = 2, d = 1 in [10], or for example in the special case of theorem 1.3.…”