Microlocal analysis of a spindle transform
James W. Webber, Sean Holman · Inverse Problems and Imaging · 2019
An analysis of the stability of the spindle transform, introduced in [16], 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. [4]. We find that the normal operator for the spindle transform belongs to a class of distibutions $ I^{p, l}(\Delta, \Lambda)+I^{p, l}(\widetilde{\Delta}, \Lambda) $ studied by Felea and Marhuenda in [4,10], 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.