# Torus computed tomography

## TL;DR

This paper introduces Torus CT, a novel 2D computed tomography method based on the geometry of the flat torus, providing new inversion formulas, regularization strategies, and promising simulation results.

## Contribution

The paper develops a new CT inversion technique using torus geometry, including formulas, regularization, and implementation details, advancing the field of tomographic reconstruction.

## Key findings

- New inversion formulas for the Radon transform on the flat torus
- Regularization strategy with stability estimates
- Successful implementation with simulated data

## 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.