Microlocal Analysis of Generalized Radon Transforms from Scattering Tomography

Abstract

Here we present a novel microlocal analysis of generalized Radon transforms which describe the integrals of $L^2$ functions of compact support over surfaces of revolution of $C^{\infty}$ curves $q$. We show that the Radon transforms are elliptic Fourier integral operators (FIO) and provide an analysis of the left projections $\Pi_L$. Our main theorem shows that $\Pi_L$ satisfies the semiglobal Bolker assumption if and only if $g=q'/q$ is an immersion. An analysis of the visible singularities is presented, after which we derive novel Sobolev smoothness estimates for the generalized Radon FIO. Our theory has specific applications of interest in Emission Compton Scattering Tomography (ECST) and Bragg Scattering Tomography (BST). We show that the ECST and BST integration curves satisfy the semiglobal Bolker assumption and provide simulated reconstructions from ECST and BST data. Additionally, we give example “sinusoidal" integration curves which do not satisfy Bolker and provide simulations of the image artifacts. The observed artifacts in reconstruction are shown to align exactly with our predictions.