# Three dimensional Compton scattering tomography

## TL;DR

This paper introduces a novel 3D X-ray Compton imaging method using a new acquisition geometry, proving injectivity of associated transforms and demonstrating its effectiveness through simulated reconstructions with noise and error modeling.

## Contribution

It develops a new 3D inverse problem framework for electron density reconstruction from spindle tori integrals and proves injectivity of related transforms, including polychromatic source cases.

## Key findings

- Injectivity of the generalized spindle torus transform is established.
- Explicit inversion formulas for harmonic coefficients are derived.
- Simulated reconstructions demonstrate robustness to noise and systematic errors.

## Abstract

We propose a new acquisition geometry for electron density reconstruction in three dimensional X-ray Compton imaging using a monochromatic source. This leads us to a new three dimensional inverse problem where we aim to reconstruct a real valued function $f$ (the electron density) from its integrals over spindle tori. We prove injectivity of a generalized spindle torus transform on the set of smooth functions compactly supported on a hollow ball. This is obtained through the explicit inversion of a class of Volterra integral operators, whose solutions give us an expression for the harmonic coefficients of $f$. The polychromatic source case is later considered, and we prove injectivity of a new spindle interior transform, apple transform and apple interior transform on the set of smooth functions compactly supported on a hollow ball. A possible physical model is suggested for both source types. We also provide simulated density reconstructions with varying levels of added pseudo random noise and model the systematic error due to the attenuation of the incoming and scattered rays in our simulation.