2017/06/10 by James W. Webber, Webber, James, Sean Holman +1 · 1 citation
Earth and Planetary Sciences · Mathematics · Medicine · #FOS: Mathematics #Functional Analysis (math.FA) #Medical Imaging Techniques and Applications #Numerical methods in inverse problems #Seismic Imaging and Inversion Techniques
paper · pdf · doi:10.48550/arxiv.1706.03168
openalex publication_date 2017/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Ip,l(Δ∪\widetildeΔ,Λ) 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 \widetildeΔ is reflection through the origin, and Λ 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 Λ and show how the filter we derived can be applied to reduce the strength of the artefact.