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

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.