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

Generalized Abel equations and applications to translation invariant Radon transforms