Surface of revolution Radon transforms with centers on generalized surfaces in $\mathbb{R}^n$ (2312.15635v1)

Abstract: We present a novel analysis of a Radon transform, $R$, which maps an $L^2$ function of compact support to its integrals over smooth surfaces of revolution with centers on an embedded hypersurface in $\mathbb{R}^n$. Using microlocal analysis, we derive necessary and sufficient conditions relating to $R$ for the Bolker condition to hold, which has implications regarding the existence and location of image artifacts. We present a general inversion framework based on Volterra equation theory and known results on the spherical Radon transform, and we prove injectivity results for $R$. Several example applications of our theory are discussed in the context of, e.g., Compton Scatter Tomography (CST) and Ultrasound Reflection Tomography (URT). In addition, using the proposed inversion framework, we validate our microlocal theory via simulation, and present simulated image reconstructions of image phantoms with added noise.