# Microlocal analysis of a Compton tomography problem

## TL;DR

This paper provides a microlocal analysis of a new toric section transform in Compton tomography, revealing image artefacts, explicit artefact expressions, and conditions for injectivity, supported by simulations and reconstructions.

## Contribution

It introduces a microlocal analysis of the toric section transform, showing artefacts in reconstructions and proving injectivity for certain functions, advancing understanding of Compton tomography imaging.

## Key findings

- The canonical relation of the transform is 2--1, leading to artefacts.
- Explicit expressions for artefacts are derived and validated by simulations.
- Injectivity is proven for functions supported inside the open unit ball.

## Abstract

Here we present a novel microlocal analysis of a new toric section transform which describes a two dimensional image reconstruction problem in Compton scattering tomography and airport baggage screening. By an analysis of two separate limited data problems for the circle transform and using microlocal analysis, we show that the canonical relation of the toric section transform is 2--1. This implies that there are image artefacts in the filtered backprojection reconstruction. We provide explicit expressions for the expected artefacts and demonstrate these by simulations. In addition, we prove injectivity of the forward operator for $L^\infty$ functions supported inside the open unit ball. We present reconstructions from simulated data using a discrete approach and several regularizers with varying levels of added pseudo-random noise.