# 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:
 [1] J. R. Driscoll and D. M. Healy, Computing Fourier transforms and convolutions on the 2-sphere, Advances in Applied Mathematics, 15 (1994), 202-250.  doi: 10.1006/aama.1994.1008.  Google Scholar [2] J. J. Duistermaat, Fourier Integral Operators, volume 130. Springer Science & Business Media, 1996.  Google Scholar [3] R. Felea, Composition of Fourier integral operators with fold and blowdown singularities, Communications in Partial Differential Equations, 30 (2005), 1717-1740.  doi: 10.1080/03605300500299968.  Google Scholar [4] R. Felea, R. Gaburro and C. J. Nolan, Microlocal analysis of SAR imaging of a dynamic reflectivity function, SIAM Journal on Mathematical Analysis, 45 (2013), 2767-2789.  doi: 10.1137/120873571.  Google Scholar [5] A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, Journal of Functional Analysis, 89 (1990), 202-232.  doi: 10.1016/0022-1236(90)90011-9.  Google Scholar [6] V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. J, 48 (1981), 251-267.  doi: 10.1215/S0012-7094-81-04814-6.  Google Scholar [7] L. Hörmander, The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators. Corrected reprint of the 1985 original, Springer, 1994. Google Scholar [8] L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution theory and Fourier analysis. Reprint of the second (1990) edition, Springer, Berlin, 2003. doi: 10.1007/978-3-642-61497-2.  Google Scholar [9] L. Hörmander, The Analysis of Linear Partial Differential Operators III, Springer, 2007. doi: 10.1007/978-3-540-49938-1.  Google Scholar [10] F. Marhuenda, Microlocal analysis of some isospectral deformations, Transactions of the American Mathematical Society, 343 (1994), 245-275.  doi: 10.1090/S0002-9947-1994-1181185-0.  Google Scholar [11] F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001. doi: 10.1137/1.9780898719284.  Google Scholar [12] M. K. Nguyen and T. T. Truong, Inversion of a new circular-arc radon transform for Compton scattering tomography, Inverse Problems, 26 (2010), 099802, 1 p. doi: 10.1088/0266-5611/26/9/099802.  Google Scholar [13] S. J. Norton, Compton scattering tomography, Journal of Applied Physics, 76 (1994), 2007-2015.  doi: 10.1063/1.357668.  Google Scholar [14] V. P. Palamodov, An analytic reconstruction for the Compton scattering tomography in a plane, Inverse Problems, 27 (2011), 125004, 8pp. doi: 10.1088/0266-5611/27/12/125004.  Google Scholar [15] R. T. Seeley, Spherical harmonics, The American Mathematical Monthly, 73 (1966), 115-121.  doi: 10.1080/00029890.1966.11970927.  Google Scholar [16] J. Webber and W. Lionheart, Three dimensional Compton scattering tomography, Inverse Problems, 34 (2018), arXiv: 1704.03378. doi: 10.1088/1361-6420/aac51e.  Google Scholar

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##### References:
 [1] J. R. Driscoll and D. M. Healy, Computing Fourier transforms and convolutions on the 2-sphere, Advances in Applied Mathematics, 15 (1994), 202-250.  doi: 10.1006/aama.1994.1008.  Google Scholar [2] J. J. Duistermaat, Fourier Integral Operators, volume 130. Springer Science & Business Media, 1996.  Google Scholar [3] R. Felea, Composition of Fourier integral operators with fold and blowdown singularities, Communications in Partial Differential Equations, 30 (2005), 1717-1740.  doi: 10.1080/03605300500299968.  Google Scholar [4] R. Felea, R. Gaburro and C. J. Nolan, Microlocal analysis of SAR imaging of a dynamic reflectivity function, SIAM Journal on Mathematical Analysis, 45 (2013), 2767-2789.  doi: 10.1137/120873571.  Google Scholar [5] A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, Journal of Functional Analysis, 89 (1990), 202-232.  doi: 10.1016/0022-1236(90)90011-9.  Google Scholar [6] V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. J, 48 (1981), 251-267.  doi: 10.1215/S0012-7094-81-04814-6.  Google Scholar [7] L. Hörmander, The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators. Corrected reprint of the 1985 original, Springer, 1994. Google Scholar [8] L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution theory and Fourier analysis. Reprint of the second (1990) edition, Springer, Berlin, 2003. doi: 10.1007/978-3-642-61497-2.  Google Scholar [9] L. Hörmander, The Analysis of Linear Partial Differential Operators III, Springer, 2007. doi: 10.1007/978-3-540-49938-1.  Google Scholar [10] F. Marhuenda, Microlocal analysis of some isospectral deformations, Transactions of the American Mathematical Society, 343 (1994), 245-275.  doi: 10.1090/S0002-9947-1994-1181185-0.  Google Scholar [11] F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001. doi: 10.1137/1.9780898719284.  Google Scholar [12] M. K. Nguyen and T. T. Truong, Inversion of a new circular-arc radon transform for Compton scattering tomography, Inverse Problems, 26 (2010), 099802, 1 p. doi: 10.1088/0266-5611/26/9/099802.  Google Scholar [13] S. J. Norton, Compton scattering tomography, Journal of Applied Physics, 76 (1994), 2007-2015.  doi: 10.1063/1.357668.  Google Scholar [14] V. P. Palamodov, An analytic reconstruction for the Compton scattering tomography in a plane, Inverse Problems, 27 (2011), 125004, 8pp. doi: 10.1088/0266-5611/27/12/125004.  Google Scholar [15] R. T. Seeley, Spherical harmonics, The American Mathematical Monthly, 73 (1966), 115-121.  doi: 10.1080/00029890.1966.11970927.  Google Scholar [16] J. Webber and W. Lionheart, Three dimensional Compton scattering tomography, Inverse Problems, 34 (2018), arXiv: 1704.03378. doi: 10.1088/1361-6420/aac51e.  Google Scholar
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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