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.

Broken Ray Transform for Twisted Geodesics on Surfaces with a Reflecting Obstacle

“…Other geometric results include boundary determination from a broken ray transform [32] and a reflection approach using strong symmetry assumptions [34], for example letting to solve the broken ray transforms on flat boxes over closed billiard trajectories [29,33]. Numerical reconstruction algorithms and stability for the mentioned problem on the flat boxes would follow directly from [31,62]. Artifacts appearing in the inversion of a broken ray transform was studied recently in the flat geometry [72].…”

On the range of the planar $X$-ray transform on the Fourier lattice of the torus

“…In here we give yet another characterization of the range of the classical X-ray transform in terms of the Fourier coefficients of integrable functions on the torus, where the lines passing through the support of f are parameterized by coordinates on a torus. Although Xf is a function on the torus, our problem differs from the one in [13], where for a given direction (of rational slope) the integration takes place over a finite union of parallel segments in the unit disc.…”

On the range of the $X$-ray transform of symmetric tensors compactly supported in the plane

“…This novel point of view allowed to establish the missing connection between the result in [38] and the classical GGHL characterization. Although Xf is a function on the torus, this problem differs from the one in [15], where for a given direction (of rational slope) the integration takes place over a finite union of parallel segments in the unit disc.…”