American Institute of Mathematical Sciences

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

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

Yang Zhang ,

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.

Keywords: Broken ray transform, artifacts, recovery of singularities.
Mathematics Subject Classification: Primary: 35R30, 44A12; Secondary: 65R32.
Citation: Yang Zhang. Artifacts in the inversion of the broken ray transform in the plane. Inverse Problems & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

B. Frigyik, P. 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.  Google Scholar

[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.  Google Scholar

[10]

M. Haltmeier, S. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

F. Monard, P. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

S. Park, An introduction to dynamical billiards. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[40]

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

[41]

W. Zhang, D. Zhu, M. 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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

B. Frigyik, P. 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.  Google Scholar

[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.  Google Scholar

[10]

M. Haltmeier, S. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

F. Monard, P. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

S. Park, An introduction to dynamical billiards. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[40]

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

[41]

W. Zhang, D. Zhu, M. 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.  Google Scholar

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 Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The small neighborhood $ U_k $ and $ (x_k, \xi^k) $, for $ k = 1, 2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  A sketch of a broken ray reflected on a smooth boundary and the notation
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Two broken rays intersect when $ \alpha_2 $ increases as $ \alpha_1 $ increases
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Artifacts and caustics. Form left to right: $ f $, $ {{B}}^*\Lambda {{B}} f $, and caustics caused by reflected light
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Local reconstruction by Landweber iteration
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Inside a circular mirror, a sequence of broken rays and conjugate points on them
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

Figure 14.  Left to right: true $ f $, backprojection $ f^{(1)} $, $ f^{(100)} $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Mark Hubenthal. The broken ray transform in $n$ dimensions with flat reflecting boundary. Inverse Problems & Imaging, 2015, 9 (1) : 143-161. doi: 10.3934/ipi.2015.9.143
[2]

Gareth Ainsworth. The attenuated magnetic ray transform on surfaces. Inverse Problems & Imaging, 2013, 7 (1) : 27-46. doi: 10.3934/ipi.2013.7.27
[3]

Gareth Ainsworth. The magnetic ray transform on Anosov surfaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1801-1816. doi: 10.3934/dcds.2015.35.1801
[4]

Dan Jane, Gabriel P. Paternain. On the injectivity of the X-ray transform for Anosov thermostats. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 471-487. doi: 10.3934/dcds.2009.24.471
[5]

Yiran Wang. Parametrices for the light ray transform on Minkowski spacetime. Inverse Problems & Imaging, 2018, 12 (1) : 229-237. doi: 10.3934/ipi.2018009
[6]

Gareth Ainsworth, Yernat M. Assylbekov. On the range of the attenuated magnetic ray transform for connections and Higgs fields. Inverse Problems & Imaging, 2015, 9 (2) : 317-335. doi: 10.3934/ipi.2015.9.317
[7]

Siamak RabieniaHaratbar. Support theorem for the Light-Ray transform of vector fields on Minkowski spaces. Inverse Problems & Imaging, 2018, 12 (2) : 293-314. doi: 10.3934/ipi.2018013
[8]

François Rouvière. X-ray transform on Damek-Ricci spaces. Inverse Problems & Imaging, 2010, 4 (4) : 713-720. doi: 10.3934/ipi.2010.4.713
[9]

Jan Boman. Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform. Inverse Problems & Imaging, 2010, 4 (4) : 619-630. doi: 10.3934/ipi.2010.4.619
[10]

Venkateswaran P. Krishnan, Plamen Stefanov. A support theorem for the geodesic ray transform of symmetric tensor fields. Inverse Problems & Imaging, 2009, 3 (3) : 453-464. doi: 10.3934/ipi.2009.3.453
[11]

Lassi Päivärinta, Valery Serov. Recovery of jumps and singularities in the multidimensional Schrodinger operator from limited data. Inverse Problems & Imaging, 2007, 1 (3) : 525-535. doi: 10.3934/ipi.2007.1.525
[12]

Aleksander Denisiuk. On range condition of the tensor x-ray transform in $ \mathbb R^n $. Inverse Problems & Imaging, 2020, 14 (3) : 423-435. doi: 10.3934/ipi.2020020
[13]

Dominique Zosso, Jing An, James Stevick, Nicholas Takaki, Morgan Weiss, Liane S. Slaughter, Huan H. Cao, Paul S. Weiss, Andrea L. Bertozzi. Image segmentation with dynamic artifacts detection and bias correction. Inverse Problems & Imaging, 2017, 11 (3) : 577-600. doi: 10.3934/ipi.2017027
[14]

Venkateswaran P. Krishnan, Ramesh Manna, Suman Kumar Sahoo, Vladimir A. Sharafutdinov. Momentum ray transforms. Inverse Problems & Imaging, 2019, 13 (3) : 679-701. doi: 10.3934/ipi.2019031
[15]

Sara Barile, Addolorata Salvatore. Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Conference Publications, 2013, 2013 (special) : 41-49. doi: 10.3934/proc.2013.2013.41
[16]

Nicholas Hoell, Guillaume Bal. Ray transforms on a conformal class of curves. Inverse Problems & Imaging, 2014, 8 (1) : 103-125. doi: 10.3934/ipi.2014.8.103
[17]

Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006
[18]

Doǧan Çömez. The modulated ergodic Hilbert transform. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 325-336. doi: 10.3934/dcdss.2009.2.325
[19]

Sean Holman, Plamen Stefanov. The weighted Doppler transform. Inverse Problems & Imaging, 2010, 4 (1) : 111-130. doi: 10.3934/ipi.2010.4.111
[20]

James W. Webber, Sean Holman. Microlocal analysis of a spindle transform. Inverse Problems & Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (155)
  • HTML views (159)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences