Microlocal analysis of a spindle transform

James W. Webber; Sean Holman

doi:10.3934/ipi.2019013

Article Contents

2019, Volume 13, Issue 2: 231-261. Doi: 10.3934/ipi.2019013

This issue Previous Article Next Article On finding the surface admittance of an obstacle via the time domain enclosure method

Microlocal analysis of a spindle transform

James W. Webber^1, and
Sean Holman^2,

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

Received: May 31, 2017

Revised: September 30, 2017

Published: March 2019

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).

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Abstract

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.

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Mathematics Subject Classification: Primary: 35A27; Secondary: 35R30.

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Figure 1. 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

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Figure 2. Small bead

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Figure 3. Bead reconstruction by backprojection

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Figure 4. Bead reconstruction by filtered backprojection, with $ L = 25 $ components

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Figure 5. Bead reconstruction by backprojection, truncating the data to $ L = 25 $ components

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Figure 6. Layered spherical shell segment phantom, centred at the origin

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Figure 7. Layered plane phantom

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Figure 8. Layered spherical shell segment CGLS reconstruction

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Figure 9. Layered spherical shell segment CGLS reconstruction, with $ Q^{\frac{1}{2}} $ used as a pre–conditioner and no added Tikhonov regularisation

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Figure 10. Layered plane reconstruction by CGLS

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Figure 11. Layered plane reconstruction by Landweber iteration

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Figure 12. Spherical shell reconstruction by Landweber iteration

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Figure 13. Spherical shell reconstruction by Landweber iteration, with $ Q^{\frac{1}{2}} $ used as a pre–conditioner

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