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Here we present a novel microlocal analysis of generalized Radon transformswhich describe the integrals of $L^2$ functions of compact support oversurfaces of revolution of $C^{\infty}$ curves $q$. We show that the Radontransforms are elliptic Fourier Integral Operators (FIO) and provide ananalysis of the left projections $\Pi_L$. Our main theorem shows that $\Pi_L$satisfies the semi-global Bolker assumption if and only if $g=q'/q$ is animmersion. An analysis of the visible singularities is presented, after whichwe derive novel Sobolev smoothness estimates for the Radon FIO. Our theory hasspecific applications of interest in Compton Scattering Tomography (CST) andBragg Scattering Tomography (BST). We show that the CST and BST integrationcurves satisfy the Bolker assumption and provide simulated reconstructions fromCST and BST data. Additionally we give example "sinusoidal" integration curveswhich do not satisfy Bolker and provide simulations of the image artefacts. Theobserved artefacts in reconstruction are shown to align exactly with ourpredictions.

Microlocal analysis of generalized Radon transforms from scattering tomography