Microlocal analysis of a spindle transform

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.