Modified Radon transform inversion using moments

TL;DR

This paper introduces a modified Radon transform using convolution with a mollifier, derives its inversion formula, and demonstrates its effectiveness in reconstructing images from moments with numerical validation.

Contribution

It presents a novel modified Radon transform and its inversion method, enhancing image reconstruction from moments and providing theoretical and numerical validation.

Findings

The MRT provides a uniform approximation to the original density.

Numerical reconstructions agree with theoretical convergence results.

The method improves reconstruction from fewer projections and suppresses noise.

Abstract

Moment methods to reconstruct images from their Radon transforms are both natural and useful. They can be used to suppress noise or other spurious effects and can lead to highly efficient reconstructions from relatively few projections. We establish a modified Radon transform (MRT) via convolution with a mollifier and obtain its inversion formula. The relationship of the moments of the Radon transform and the moments of its modified Radon transform is derived and MRT data is used to provide a uniform approximation to the original density function. The reconstruction algorithm is implemented, and a simple density function is reconstructed from moments of its modified Radon transform. Numerical convergence of this reconstruction is shown to agree with the derived theoretical results.