# 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. [2] J. J. Duistermaat, Fourier Integral Operators, volume 130. Springer Science & Business Media, 1996. [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. [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. [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. [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. [7] L. Hörmander, The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators. Corrected reprint of the 1985 original, Springer, 1994. [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. [9] L. Hörmander, The Analysis of Linear Partial Differential Operators III, Springer, 2007. doi: 10.1007/978-3-540-49938-1. [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. [11] F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001. doi: 10.1137/1.9780898719284. [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. [13] S. J. Norton, Compton scattering tomography, Journal of Applied Physics, 76 (1994), 2007-2015. doi: 10.1063/1.357668. [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. [15] R. T. Seeley, Spherical harmonics, The American Mathematical Monthly, 73 (1966), 115-121. doi: 10.1080/00029890.1966.11970927. [16] J. Webber and W. Lionheart, Three dimensional Compton scattering tomography, Inverse Problems, 34 (2018), arXiv: 1704.03378. doi: 10.1088/1361-6420/aac51e.

show all references

##### 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. [2] J. J. Duistermaat, Fourier Integral Operators, volume 130. Springer Science & Business Media, 1996. [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. [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. [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. [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. [7] L. Hörmander, The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators. Corrected reprint of the 1985 original, Springer, 1994. [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. [9] L. Hörmander, The Analysis of Linear Partial Differential Operators III, Springer, 2007. doi: 10.1007/978-3-540-49938-1. [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. [11] F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001. doi: 10.1137/1.9780898719284. [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. [13] S. J. Norton, Compton scattering tomography, Journal of Applied Physics, 76 (1994), 2007-2015. doi: 10.1063/1.357668. [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. [15] R. T. Seeley, Spherical harmonics, The American Mathematical Monthly, 73 (1966), 115-121. doi: 10.1080/00029890.1966.11970927. [16] J. Webber and W. Lionheart, Three dimensional Compton scattering tomography, Inverse Problems, 34 (2018), arXiv: 1704.03378. doi: 10.1088/1361-6420/aac51e.
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 backprojection
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
 [1] Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure & Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1 [2] Kanghui Guo and Demetrio Labate. Sparse shearlet representation of Fourier integral operators. Electronic Research Announcements, 2007, 14: 7-19. doi: 10.3934/era.2007.14.7 [3] Aki Pulkkinen, Ville Kolehmainen, Jari P. Kaipio, Benjamin T. Cox, Simon R. Arridge, Tanja Tarvainen. Approximate marginalization of unknown scattering in quantitative photoacoustic tomography. Inverse Problems & Imaging, 2014, 8 (3) : 811-829. doi: 10.3934/ipi.2014.8.811 [4] John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181 [5] Meghdoot Mozumder, Tanja Tarvainen, Simon Arridge, Jari P. Kaipio, Cosimo D'Andrea, Ville Kolehmainen. Approximate marginalization of absorption and scattering in fluorescence diffuse optical tomography. Inverse Problems & Imaging, 2016, 10 (1) : 227-246. doi: 10.3934/ipi.2016.10.227 [6] Gary Froyland, Cecilia González-Tokman, Anthony Quas. Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools. Journal of Computational Dynamics, 2014, 1 (2) : 249-278. doi: 10.3934/jcd.2014.1.249 [7] Dorota Bors, Andrzej Skowron, Stanisław Walczak. Systems described by Volterra type integral operators. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2401-2416. doi: 10.3934/dcdsb.2014.19.2401 [8] Patricio Felmer, Alexander Quaas. Fundamental solutions for a class of Isaacs integral operators. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 493-508. doi: 10.3934/dcds.2011.30.493 [9] Hermann Brunner. On Volterra integral operators with highly oscillatory kernels. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 915-929. doi: 10.3934/dcds.2014.34.915 [10] Beatrice Bugert, Gunther Schmidt. Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 435-473. doi: 10.3934/dcdss.2015.8.435 [11] Daniela Calvetti, Erkki Somersalo. Microlocal sequential regularization in imaging. Inverse Problems & Imaging, 2007, 1 (1) : 1-11. doi: 10.3934/ipi.2007.1.1 [12] Kirill D. Cherednichenko, Alexander V. Kiselev, Luis O. Silva. Functional model for extensions of symmetric operators and applications to scattering theory. Networks & Heterogeneous Media, 2018, 13 (2) : 191-215. doi: 10.3934/nhm.2018009 [13] Alexei Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators. Inverse Problems & Imaging, 2009, 3 (1) : 139-149. doi: 10.3934/ipi.2009.3.139 [14] Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248 [15] Pavel Krejčí, Harbir Lamba, Sergey Melnik, Dmitrii Rachinskii. Kurzweil integral representation of interacting Prandtl-Ishlinskii operators. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2949-2965. doi: 10.3934/dcdsb.2015.20.2949 [16] Congming Li, Jisun Lim. The singularity analysis of solutions to some integral equations. Communications on Pure & Applied Analysis, 2007, 6 (2) : 453-464. doi: 10.3934/cpaa.2007.6.453 [17] Sarah Constantin, Robert S. Strichartz, Miles Wheeler. Analysis of the Laplacian and spectral operators on the Vicsek set. Communications on Pure & Applied Analysis, 2011, 10 (1) : 1-44. doi: 10.3934/cpaa.2011.10.1 [18] Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems & Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139 [19] Daniel Bouche, Youngjoon Hong, Chang-Yeol Jung. Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1159-1181. doi: 10.3934/dcds.2017048 [20] Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41

2017 Impact Factor: 1.465