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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2019, 13(2): 231-261.
doi: 10.3934/ipi.2019013
Microlocal analysis of a spindle transform
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An analysis of the stability of the spindle transform, introduced in [
Keywords:
Microlocal analysis,
Fourier integral operators,
Compton scattering,
tomography,
artefact reduction.
Mathematics Subject Classification:
Primary: 35A27; Secondary: 35R30.
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. 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. |
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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