# American Institute of Mathematical Sciences

April  2019, 13(2): 231-261. doi: 10.3934/ipi.2019013

## Microlocal analysis of a spindle transform

 1 Halligan Hall, 161 College Ave, Medford MA 02144, USA 2 Alan Turing Building, Oxford Road, Manchester, Greater Manchester M13 9PL, UK

Received  June 2017 Revised  October 2017 Published  January 2019

Fund Project: The first author is supported by Engineering and Physical Sciences Research Council and Rapiscan systems, CASE studentship
The second author is supported by Engineering and Physical Sciences Research Council (EP/M016773/1)

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.

Citation: James W. Webber, Sean Holman. Microlocal analysis of a spindle transform. Inverse Problems & Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013
##### References:

show all references

##### References:
A spindle torus with axis of rotation $\theta$, tube centre offset $r$ and tube radius $\sqrt{1+r^2}$. The distance between the origin and either of the points where the torus self intersects is 1
Bead reconstruction by filtered backprojection, with $L = 25$ components
Bead reconstruction by backprojection, truncating the data to $L = 25$ components
Layered spherical shell segment phantom, centred at the origin
Layered plane phantom
Layered spherical shell segment CGLS reconstruction
Layered spherical shell segment CGLS reconstruction, with $Q^{\frac{1}{2}}$ used as a pre–conditioner and no added Tikhonov regularisation
Layered plane reconstruction by CGLS
Layered plane reconstruction by Landweber iteration
Spherical shell reconstruction by Landweber iteration
Spherical shell reconstruction by Landweber iteration, with $Q^{\frac{1}{2}}$ used as a pre–conditioner
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