Microlocal analysis of generalized Radon transforms from scattering tomography

TL;DR

This paper provides a microlocal analysis of generalized Radon transforms related to scattering tomography, establishing conditions for ellipticity and the Bolker assumption, and demonstrating applications in CST and BST with simulations.

Contribution

It introduces a novel microlocal framework for analyzing Radon transforms in scattering tomography, including new Sobolev estimates and conditions for the Bolker assumption.

Findings

Radon transforms are elliptic Fourier Integral Operators.

The Bolker assumption holds if and only if g=q'/q is an immersion.

Simulated reconstructions confirm theoretical predictions about artifacts.

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 semi-global 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 Radon FIO. Our theory has specific applications of interest in Compton Scattering Tomography (CST) and Bragg Scattering Tomography (BST). We show that the CST and BST integration curves satisfy the Bolker assumption and provide simulated reconstructions from CST and BST data. Additionally we give example "sinusoidal" integration curves which do not satisfy Bolker and provide simulations of the image artefacts. The observed artefacts in reconstruction are shown to align exactly with our predictions.