Generalized Abel equations and applications to translation invariant Radon transforms

Generalized Abel equations have been employed in the recent literature to invert Radon transforms which arise in a number of important imaging applications, including Compton Scatter Tomography (CST), Ultrasound Reflection Tomography (URT), and X-ray CT. In this paper, we present novel injectivity results and inversion methods for Generalized Abel operators. We apply our theory to a new Radon transform, $\mathcal{R}_j$, of interest in URT, which integrates a square integrable function of compact support, $f$, over ellipsoid and hyperboloid surfaces with centers on a plane. Using our newly established theory on generalized Abel equations, we show that $\mathcal{R}_j$ is injective and provide an inversion method based on Neumann series. In addition, using algebraic methods, we present image phantom reconstructions from $\mathcal{R}_jf$ data with added pseudo random noise.