February  2020, 14(1): 1-26. doi: 10.3934/ipi.2019061

Artifacts in the inversion of the broken ray transform in the plane

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

* Corresponding author: Yang Zhang

Received  July 2018 Revised  June 2019 Published  November 2019

Fund Project: The first author is supported by NSF grant DMS-1600327.

We study the integral transform over a general family of broken rays in $ \mathbb{R}^2 $. One example of the broken rays is the family of rays reflected from a curved boundary once. There is a natural notion of conjugate points for broken rays. If there are conjugate points, we show that the singularities conormal to the broken rays cannot be recovered from local data and therefore artifacts arise in the reconstruction. As for global data, more singularities might be recoverable. We apply these conclusions to two examples, the V-line transform and the parallel ray transform. In each example, a detailed discussion of the local and global recovery of singularities is given and we perform numerical experiments to illustrate the results.

Citation: Yang Zhang. Artifacts in the inversion of the broken ray transform in the plane. Inverse Problems and Imaging, 2020, 14 (1) : 1-26. doi: 10.3934/ipi.2019061
References:
[1]

G. Ambartsoumian, Inversion of the V-line Radon transform in a disc and its applications in imaging, Comput. Math. Appl., 64 (2012), 260-265.  doi: 10.1016/j.camwa.2012.01.059.

[2]

R. Basko, G. L. Zeng and G. T. Gullberg, Application of spherical harmonics to image reconstruction for the Compton camera, Phys. Med. Biol., 43 (1998). doi: 10.1088/0031-9155/43/4/016.

[3]

J. A. Boyle, Using rolling circles to generate caustic envelopes resulting from reflected light, Amer. Math. Monthly, 122 (2015), 452-466.  doi: 10.4169/amer.math.monthly.122.5.452.

[4]

A. Cayley, A memoir upon caustics, Philos. Trans. Royal Soc. London, 147 (1857), 273-312. 

[5]

H. S. M. Coxeter, Introduction to Geometry, John Wiley & Sons, Inc., New York-London, 1961.

[6]

D. B. Everett, J. S. Fleming, R. W. Todd and J. M. Nightingale, Gamma-radiation imaging system based on the Compton effect, Proceedings of the Institution of Electrical Engineers, 124, 1977. doi: 10.1049/piee.1977.0203.

[7]

L. Florescu, V. A. Markel and J. C. Schotland, Inversion formulas for the broken-ray Radon transform, Inverse Problems, 27 (2011), 13pp. doi: 10.1088/0266-5611/27/2/025002.

[8]

B. FrigyikP. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights, J. Geom. Anal., 18 (2008), 89-108.  doi: 10.1007/s12220-007-9007-6.

[9]

R. Gouia-Zarrad and G. Ambartsoumian, Exact inversion of the conical Radon transform with a fixed opening angle, Inverse Problems, 30 (2014), 12pp. doi: 10.1088/0266-5611/30/4/045007.

[10]

M. HaltmeierS. Moon and D. Schiefeneder, Inversion of the attenuated V-line transform with vertices on the circle, IEEE Trans. Comput. Imaging, 3 (2017), 853-863.  doi: 10.1109/TCI.2017.2669868.

[11]

S. Holman, F. Monard and P. Stefanov, The attenuated geodesic X-ray transform, Inverse Problems, 34 (2018), 26pp. doi: 10.1088/1361-6420/aab8bc.

[12]

S. Holman and G. Uhlmann, On the microlocal analysis of the geodesic X-ray transform with conjugate points, J. Differential Geom., 108 (2018), 459-494.  doi: 10.4310/jdg/1519959623.

[13]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Classics in Mathematics, Springer, Berlin, 2007. doi: 10.1007/978-3-540-49938-1.

[14]

M. Hubenthal, The broken ray transform on the square, J. Fourier Anal. Appl., 20 (2014), 1050-1082.  doi: 10.1007/s00041-014-9344-3.

[15]

M. Hubenthal, The broken ray transform in $n$ dimensions with flat reflecting boundary, Inverse Probl. Imaging, 9 (2015), 143-161.  doi: 10.3934/ipi.2015.9.143.

[16]

J. Ilmavirta, Broken ray tomography in the disc, Inverse Problems, 29 (2013), 17pp. doi: 10.1088/0266-5611/29/3/035008.

[17]

J. Ilmavirta, On the broken ray transform, preprint, arXiv: 1409.7500.

[18]

J. Ilmavirta, A reflection approach to the broken ray transform, Math. Scand., 117 (2015), 231-257.  doi: 10.7146/math.scand.a-22869.

[19]

J. Ilmavirta and M. Salo, Broken ray transform on a Riemann surface with a convex obstacle, Comm. Anal. Geom., 24 (2016), 379-408.  doi: 10.4310/CAG.2016.v24.n2.a6.

[20]

C.-Y. Jung and S. Moon, Inversion formulas for cone transforms arising in application of Compton cameras, Inverse Problems, 31 (2015), 20pp. doi: 10.1088/0266-5611/31/1/015006.

[21]

R. Krylov and A. Katsevich, Inversion of the broken ray transform in the case of energy-dependent attenuation, Phys. Med. Biol., 60 (2015), 4313-4334.  doi: 10.1088/0031-9155/60/11/4313.

[22]

P. Kuchment and F. Terzioglu, Three-dimensional image reconstruction from Compton camera data, SIAM J. Imaging Sci., 9 (2016), 1708-1725.  doi: 10.1137/16M107476X.

[23]

V. Maxim, M. Frande and R. Prost, Analytical inversion of the Compton transform using the full set of available projections, Inverse Problems, 25 (2009), 21pp. doi: 10.1088/0266-5611/25/9/095001.

[24]

F. MonardP. Stefanov and G. Uhlmann, The geodesic ray transform on Riemannian surfaces with conjugate points, Comm. Math. Phys., 337 (2015), 1491-1513.  doi: 10.1007/s00220-015-2328-6.

[25]

S. Moon, On the determination of a function from its conical Radon transform with a fixed central axis, SIAM J. Math. Anal., 48 (2016), 1833-1847.  doi: 10.1137/15M1021945.

[26]

S. Moon and M. Haltmeier, Analytic inversion of a conical Radon transform arising in application of Compton cameras on the cylinder, SIAM J. Imaging Sci., 10 (2017), 535-557.  doi: 10.1137/16M1083116.

[27]

M. Morvidone, M. K. Nguyen, T. Truong and H. Zaidi, On the V-line Radon transform and its imaging applications, IEEE International Conference on Image Processing, Hong Kong, 2010. doi: 10.1109/ICIP.2010.5653835.

[28]

S. Park, An introduction to dynamical billiards.

[29]

L. C. Parra, Reconstruction of cone-beam projections from Compton scattered data, IEEE Trans. Nuclear Science, 47 (2000), 1543-1550.  doi: 10.1109/NSSMIC.1999.845848.

[30]

D. Schiefeneder and M. Haltmeier, The Radon transform over cones with vertices on the sphere and orthogonal axes, SIAM J. Appl. Math., 77 (2017), 1335-1351.  doi: 10.1137/16M1079476.

[31]

M. Singh, An electronically collimated gamma camera for single photon emission computed tomography. Part Ⅰ: Theoretical considerations and design criteria, J. Comput. Assisted Tomography, 7 (1983), 421-427.  doi: 10.1097/00004728-198312000-00071.

[32]

B. Smith, Reconstruction methods and completeness conditions for two Compton data models, JOSA A, 22 (2005), 445-459.  doi: 10.1364/JOSAA.22.000445.

[33]

P. Stefanov and G. Uhlmann, The geodesic X-ray transform with fold caustics, Anal. PDE, 5 (2012), 219-260.  doi: 10.2140/apde.2012.5.219.

[34]

P. Stefanov and G. Uhlmann, Is a curved flight path in SAR better than a straight one?, SIAM J. Appl. Math., 73 (2013), 1596-1612.  doi: 10.1137/120882639.

[35]

P. Stefanov and Y. Yang, Multiwave tomography with reflectors: Landweber's iteration, Inverse Probl. Imaging, 11 (2017), 373-401.  doi: 10.3934/ipi.2017018.

[36]

F. Terzioglu, Some inversion formulas for the cone transform, Inverse Problems, 31 (2015), 21pp. doi: 10.1088/0266-5611/31/11/115010.

[37]

F. Terzioglu and P. Kuchment, Inversion of weighted divergent beam and cone transforms, Inverse Probl. Imaging, 11 (2017), 1071-1090.  doi: 10.3934/ipi.2017049.

[38]

R. W. ToddJ. M. Nightingale and D. B. Everett, A proposed $\gamma$ camera, Nature, 251 (1974), 132-134.  doi: 10.1038/251132a0.

[39]

T. Tomitani and M. Hirasawa, Image reconstruction from limited angle Compton camera data, Phys. Med. Biol., 47 (2002). doi: 10.1088/0031-9155/47/12/309.

[40]

F. W. Warner, The conjugate locus of a Riemannian manifold, Amer. J. Math., 87 (1965), 575-604.  doi: 10.2307/2373064.

[41]

W. ZhangD. ZhuM. Lun and C. Li, Multiple pinhole collimator based X-ray luminescence computed tomography, Biomed. Opt. Express, 7 (2016), 2506-2523.  doi: 10.1364/BOE.7.002506.

show all references

References:
[1]

G. Ambartsoumian, Inversion of the V-line Radon transform in a disc and its applications in imaging, Comput. Math. Appl., 64 (2012), 260-265.  doi: 10.1016/j.camwa.2012.01.059.

[2]

R. Basko, G. L. Zeng and G. T. Gullberg, Application of spherical harmonics to image reconstruction for the Compton camera, Phys. Med. Biol., 43 (1998). doi: 10.1088/0031-9155/43/4/016.

[3]

J. A. Boyle, Using rolling circles to generate caustic envelopes resulting from reflected light, Amer. Math. Monthly, 122 (2015), 452-466.  doi: 10.4169/amer.math.monthly.122.5.452.

[4]

A. Cayley, A memoir upon caustics, Philos. Trans. Royal Soc. London, 147 (1857), 273-312. 

[5]

H. S. M. Coxeter, Introduction to Geometry, John Wiley & Sons, Inc., New York-London, 1961.

[6]

D. B. Everett, J. S. Fleming, R. W. Todd and J. M. Nightingale, Gamma-radiation imaging system based on the Compton effect, Proceedings of the Institution of Electrical Engineers, 124, 1977. doi: 10.1049/piee.1977.0203.

[7]

L. Florescu, V. A. Markel and J. C. Schotland, Inversion formulas for the broken-ray Radon transform, Inverse Problems, 27 (2011), 13pp. doi: 10.1088/0266-5611/27/2/025002.

[8]

B. FrigyikP. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights, J. Geom. Anal., 18 (2008), 89-108.  doi: 10.1007/s12220-007-9007-6.

[9]

R. Gouia-Zarrad and G. Ambartsoumian, Exact inversion of the conical Radon transform with a fixed opening angle, Inverse Problems, 30 (2014), 12pp. doi: 10.1088/0266-5611/30/4/045007.

[10]

M. HaltmeierS. Moon and D. Schiefeneder, Inversion of the attenuated V-line transform with vertices on the circle, IEEE Trans. Comput. Imaging, 3 (2017), 853-863.  doi: 10.1109/TCI.2017.2669868.

[11]

S. Holman, F. Monard and P. Stefanov, The attenuated geodesic X-ray transform, Inverse Problems, 34 (2018), 26pp. doi: 10.1088/1361-6420/aab8bc.

[12]

S. Holman and G. Uhlmann, On the microlocal analysis of the geodesic X-ray transform with conjugate points, J. Differential Geom., 108 (2018), 459-494.  doi: 10.4310/jdg/1519959623.

[13]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Classics in Mathematics, Springer, Berlin, 2007. doi: 10.1007/978-3-540-49938-1.

[14]

M. Hubenthal, The broken ray transform on the square, J. Fourier Anal. Appl., 20 (2014), 1050-1082.  doi: 10.1007/s00041-014-9344-3.

[15]

M. Hubenthal, The broken ray transform in $n$ dimensions with flat reflecting boundary, Inverse Probl. Imaging, 9 (2015), 143-161.  doi: 10.3934/ipi.2015.9.143.

[16]

J. Ilmavirta, Broken ray tomography in the disc, Inverse Problems, 29 (2013), 17pp. doi: 10.1088/0266-5611/29/3/035008.

[17]

J. Ilmavirta, On the broken ray transform, preprint, arXiv: 1409.7500.

[18]

J. Ilmavirta, A reflection approach to the broken ray transform, Math. Scand., 117 (2015), 231-257.  doi: 10.7146/math.scand.a-22869.

[19]

J. Ilmavirta and M. Salo, Broken ray transform on a Riemann surface with a convex obstacle, Comm. Anal. Geom., 24 (2016), 379-408.  doi: 10.4310/CAG.2016.v24.n2.a6.

[20]

C.-Y. Jung and S. Moon, Inversion formulas for cone transforms arising in application of Compton cameras, Inverse Problems, 31 (2015), 20pp. doi: 10.1088/0266-5611/31/1/015006.

[21]

R. Krylov and A. Katsevich, Inversion of the broken ray transform in the case of energy-dependent attenuation, Phys. Med. Biol., 60 (2015), 4313-4334.  doi: 10.1088/0031-9155/60/11/4313.

[22]

P. Kuchment and F. Terzioglu, Three-dimensional image reconstruction from Compton camera data, SIAM J. Imaging Sci., 9 (2016), 1708-1725.  doi: 10.1137/16M107476X.

[23]

V. Maxim, M. Frande and R. Prost, Analytical inversion of the Compton transform using the full set of available projections, Inverse Problems, 25 (2009), 21pp. doi: 10.1088/0266-5611/25/9/095001.

[24]

F. MonardP. Stefanov and G. Uhlmann, The geodesic ray transform on Riemannian surfaces with conjugate points, Comm. Math. Phys., 337 (2015), 1491-1513.  doi: 10.1007/s00220-015-2328-6.

[25]

S. Moon, On the determination of a function from its conical Radon transform with a fixed central axis, SIAM J. Math. Anal., 48 (2016), 1833-1847.  doi: 10.1137/15M1021945.

[26]

S. Moon and M. Haltmeier, Analytic inversion of a conical Radon transform arising in application of Compton cameras on the cylinder, SIAM J. Imaging Sci., 10 (2017), 535-557.  doi: 10.1137/16M1083116.

[27]

M. Morvidone, M. K. Nguyen, T. Truong and H. Zaidi, On the V-line Radon transform and its imaging applications, IEEE International Conference on Image Processing, Hong Kong, 2010. doi: 10.1109/ICIP.2010.5653835.

[28]

S. Park, An introduction to dynamical billiards.

[29]

L. C. Parra, Reconstruction of cone-beam projections from Compton scattered data, IEEE Trans. Nuclear Science, 47 (2000), 1543-1550.  doi: 10.1109/NSSMIC.1999.845848.

[30]

D. Schiefeneder and M. Haltmeier, The Radon transform over cones with vertices on the sphere and orthogonal axes, SIAM J. Appl. Math., 77 (2017), 1335-1351.  doi: 10.1137/16M1079476.

[31]

M. Singh, An electronically collimated gamma camera for single photon emission computed tomography. Part Ⅰ: Theoretical considerations and design criteria, J. Comput. Assisted Tomography, 7 (1983), 421-427.  doi: 10.1097/00004728-198312000-00071.

[32]

B. Smith, Reconstruction methods and completeness conditions for two Compton data models, JOSA A, 22 (2005), 445-459.  doi: 10.1364/JOSAA.22.000445.

[33]

P. Stefanov and G. Uhlmann, The geodesic X-ray transform with fold caustics, Anal. PDE, 5 (2012), 219-260.  doi: 10.2140/apde.2012.5.219.

[34]

P. Stefanov and G. Uhlmann, Is a curved flight path in SAR better than a straight one?, SIAM J. Appl. Math., 73 (2013), 1596-1612.  doi: 10.1137/120882639.

[35]

P. Stefanov and Y. Yang, Multiwave tomography with reflectors: Landweber's iteration, Inverse Probl. Imaging, 11 (2017), 373-401.  doi: 10.3934/ipi.2017018.

[36]

F. Terzioglu, Some inversion formulas for the cone transform, Inverse Problems, 31 (2015), 21pp. doi: 10.1088/0266-5611/31/11/115010.

[37]

F. Terzioglu and P. Kuchment, Inversion of weighted divergent beam and cone transforms, Inverse Probl. Imaging, 11 (2017), 1071-1090.  doi: 10.3934/ipi.2017049.

[38]

R. W. ToddJ. M. Nightingale and D. B. Everett, A proposed $\gamma$ camera, Nature, 251 (1974), 132-134.  doi: 10.1038/251132a0.

[39]

T. Tomitani and M. Hirasawa, Image reconstruction from limited angle Compton camera data, Phys. Med. Biol., 47 (2002). doi: 10.1088/0031-9155/47/12/309.

[40]

F. W. Warner, The conjugate locus of a Riemannian manifold, Amer. J. Math., 87 (1965), 575-604.  doi: 10.2307/2373064.

[41]

W. ZhangD. ZhuM. Lun and C. Li, Multiple pinhole collimator based X-ray luminescence computed tomography, Biomed. Opt. Express, 7 (2016), 2506-2523.  doi: 10.1364/BOE.7.002506.

Figure 1.  Left: a general broken ray, where $ l_1 $ and $ l_2 $ are related by a diffeomorphism. Right: a broken ray in the reflection case
Figure 2.  The small neighborhood $ U_k $ and $ (x_k, \xi^k) $, for $ k = 1, 2 $
Figure 3.  A sketch of a broken ray reflected on a smooth boundary and the notation
Figure 4.  Two broken rays intersect when $ \alpha_2 $ increases as $ \alpha_1 $ increases
Figure 5.  In (a) and (b), the bold part is the intersection region where the incoming rays hit there and reflect with conjugate points
Figure 6.  Artifacts and caustics. Form left to right: $ f $, $ {{B}}^*\Lambda {{B}} f $, and caustics caused by reflected light
Figure 7.  Local reconstruction by Landweber iteration
Figure 8.  Inside a circular mirror, a sequence of broken rays and conjugate points on them
Figure 9.  Reconstruction of $ f_1 $ and $ f_2 $ from global data, where $ e = \frac{\|f - f^{(100)}\|_2}{\|f\|_2} $ is the relative error
Figure 10.  Reconstruction from global data for Modified Shepp-Logan phantom $ f_3 $, where $ e = \frac{\|f - f^{(100)}\|_2}{\|f\|_2} $ is the relative error
Figure 11.  The error plot for the reconstruction of $ f_1, f_2, f_3 $ in order. The first two has the same range of color bar
Figure 12.  Reconstruction of two coherent states. Left to right: true $ f $, the envelopes (caused by trajectories that carry singularities and are reflected only once), $ f^{(100)} $ (where $ e = \frac{\|f - f^{(100)}\|_2}{\|f\|_2} $), the error
Figure 13.  Another case of radial singularities. Left to right: true $ f $, reconstruction $ f^{(100)} $, error for $ f $ with radial singularities after 100 iterations. The relative error $ e $ is defined as before
Figure 14.  Left to right: true $ f $, backprojection $ f^{(1)} $, $ f^{(100)} $
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