Radon transform with Gaussian beam: Theoretical and numerical reconstruction scheme

Abstract

The Radon transform and its various types have been studied since its introduction by Johann Radon in 1917. Since the Radon transform is an integral transform that maps a given function to its line integral, it has been studied in the field of computerized tomography, which deals with electromagnetic waves that primarily travel along straight lines, such as X-rays. However, in many laser optics applications, it is assumed that the laser beam is shaped like a Gaussian bell rather than a straight line. Therefore, in tomographic modalities using optical beams, such as optical projection tomography, images reconstructed with the inversion algorithms for the standard Radon transform are subject to a loss of quality. To address this issue, one needs to consider theoretical inversion methods for Radon transforms with Gaussian beam kernels and associated numerical reconstruction methods. In this study, we consider a Radon transform with a Gaussian beam kernel (also known as the point spread function) and show the uniqueness of the inversion of this transform. Furthermore, we provide an accurate and stable numerical reconstruction algorithm using the point spread function-sequential quadratic Hamiltonian scheme. Numerical experiments with disk and Shepp–Logan phantoms demonstrate that the proposed framework provides superior reconstructions compared to the traditional filtered back-projection algorithm.