Microlocal analysis of a spindle transform

TL;DR

This paper uses microlocal analysis to study the stability and artifacts of the spindle transform, introducing a filter to reduce image artifacts and demonstrating its effectiveness through simulations.

Contribution

It provides a microlocal framework for analyzing the spindle transform's normal operator and introduces a convolution filter to mitigate artifacts.

Findings

Normal operator is a paired Lagrangian distribution with specific singularities.

A convolution filter effectively reduces the rotation artefact in reconstructed images.

Simulations demonstrate the filter's ability to diminish artefacts.

Abstract

An analysis of the stability of the spindle transform, introduced in ("Three dimensional Compton scattering tomography" arXiv:1704.03378 [math.FA]), is presented. We do this via a microlocal approach and show that the normal operator for the spindle transform is a type of paired Lagrangian operator with "blowdown--blowdown" singularities analogous to that of a limited data synthetic aperture radar (SAR) problem studied by Felea et. al. ("Microlocal analysis of SAR imaging of a dynamic reflectivity function" SIAM 2013). We find that the normal operator for the spindle transform belongs to a class of distibutions $I^{p,l}(\Delta\cup\widetilde{\Delta},\Lambda)$ studied by Felea and Marhuenda ("Microlocal analysis of SAR imaging of a dynamic reflectivity function" SIAM 2013 and "Microlocal analysis of some isospectral deformations" Trans. Amer. Math.), where $\widetilde{\Delta}$ is reflection through the origin, and $\Lambda$ is associated to a rotation artefact. Later, we derive a filter to reduce the strength of the image artefact and show that it is of convolution type. We also provide simulated reconstructions to show the artefacts produced by $\Lambda$ and show how the filter we derived can be applied to reduce the strength of the artefact.