On a cylindrical scanning modality in three-dimensional Compton scatter tomography

TL;DR

This paper introduces a new 3D Compton Scattering Tomography scanner design, analyzing its mathematical properties and demonstrating its potential for stable image reconstruction despite noise.

Contribution

It provides the first injectivity, microlocal analysis, and stability results for a novel Radon transform in 3D CST, along with simulated reconstructions.

Findings

Proved injectivity and uniqueness of the transform.

Established the Bolker condition for the transform.

Demonstrated stable 3D image reconstructions with noise.

Abstract

We present injectivity and microlocal analyses of a new generalized Radon transform, $\mathcal{R}$, which has applications to a novel scanner design in three-dimensional Compton Scattering Tomography (CST), which we also introduce here. Using Fourier decomposition and Volterra equation theory, we prove that $\mathcal{R}$ is injective and show that the image solution is unique. Using microlocal analysis, we prove that $\mathcal{R}$ satisfies the Bolker condition, and we investigate the edge detection capabilities of $\mathcal{R}$. This has important implications regarding the stability of inversion and the amplification of measurement noise. In addition, we present simulated 3-D image reconstructions from $\mathcal{R}f$ data, where $f$ is a 3-D density, with varying levels of added Gaussian noise. This paper provides the theoretical groundwork for 3-D CST using the proposed scanner design.