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 and 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. FeleaR. 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. FeleaR. 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.

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
Figure 2.  Small bead
Figure 3.  Bead reconstruction by backprojection
Figure 4.  Bead reconstruction by filtered backprojection, with $ L = 25 $ components
Figure 5.  Bead reconstruction by backprojection, truncating the data to $ L = 25 $ components
Figure 6.  Layered spherical shell segment phantom, centred at the origin
Figure 7.  Layered plane phantom
Figure 8.  Layered spherical shell segment CGLS reconstruction
Figure 9.  Layered spherical shell segment CGLS reconstruction, with $ Q^{\frac{1}{2}} $ used as a pre–conditioner and no added Tikhonov regularisation
Figure 10.  Layered plane reconstruction by CGLS
Figure 11.  Layered plane reconstruction by Landweber iteration
Figure 12.  Spherical shell reconstruction by Landweber iteration
Figure 13.  Spherical shell reconstruction by Landweber iteration, with $ Q^{\frac{1}{2}} $ used as a pre–conditioner
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