  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.
