
-
Previous Article
Cloaking for a quasi-linear elliptic partial differential equation
- IPI Home
- This Issue
-
Next Article
Numerical method for image registration model based on optimal mass transport
Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms
University of California Santa Cruz, Department of Mathematics, 1156 High St. Santa Cruz, CA 95064, USA |
This article extends the author's past work [
References:
[1] |
G. Ainsworth,
The attenuated magnetic ray transform on surfaces, Inverse Problems and Imaging, 7 (2013), 27-46.
|
[2] |
E. V. Arbuzov, A. L. Bukhgeim and S. G. Kazantsev,
Two-dimensional tomography problems and the theory of A-analytic functions, Siberian Advances in Mathematics, 8 (1998), 1-20.
|
[3] |
Y. M. Assylbekov, F. Monard and G. Uhlmann, Inversion formulas and range characterizations for the attenuated geodesic ray transform,
Journal de Mathématiques Pures et Appliquées (to appear), arXiv: 1609.04361. |
[4] |
G. Bal,
On the attenuated radon transform with full and partial measurements, Inverse Problems, 20 (2004), 399-418.
|
[5] |
G. Bal and A. Tamasan,
Inverse source problem in transport equations, SIAM J. Math. Anal., 39 (2007), 57-76.
|
[6] |
J. Boman and J.-O. Strömberg,
Novikov's inversion formula for the attenuated radon transform--a new approach, J. Geom. Anal., 14 (2004), 185-198.
|
[7] |
D. Finch, The attenuated X-ray transform: Recent developments, Inside Out, Inverse Problems and Applications, 47-66, Math. Sci. Res. Inst. Publ., 47, Cambridge Univ. Press, Cambridge, 2003. |
[8] |
S. Holman and P. Stefanov,
The weighted doppler transform, Inverse Problems and Imaging, 4 (2010), 111-130.
|
[9] |
S. G. Kazantsev and A. A. Bukhgeim,
Inversion of the scalar and vector attenuated x-ray transforms in a unit disc, J. Inv. Ill-Posed Problems, 15 (2007), 735-765.
|
[10] |
S. G. Kazantsev and A. A. Bukhgeim,
Singular value decomposition for the 2d fan-beam radon transform of tensor fields, Journal of Inverse and Ill-posed Problems, 12 (2004), 245-278.
|
[11] |
F. Monard,
Efficient tensor tomography in fan-beam coordinates, Inverse Probl. Imaging, 10 (2016), 433-459.
|
[12] |
F. Monard,
Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces, SIAM J. Math. Anal., 48 (2016), 1155-1177.
|
[13] |
F. Monard and G. P. Paternain, The geodesic X-ray transform with a $GL(n, \mathbb{C})$
-connection,
to appear in Journal of Geometric Analysis, arXiv: 1610.09571. |
[14] |
F. Natterer,
Inversion of the attenuated radon transform, Inverse Problems, 17 (2001), 113-119.
|
[15] |
F. Natterer,
The Mathematics of Computerized Tomography, SIAM, 2001. |
[16] |
R. G. Novikov, An inversion formula for the attenuated x-ray transformation, Ark. Math.,
40 (2002), 145-167, (Rapport de recherche 00/05-3 Université de Nantes, Laboratoire de
Mathématiques). |
[17] |
G. Paternain,
Inside Out II, chapter Inverse problems for connections, 2012. |
[18] |
G. Paternain, M. Salo and G. Uhlmann,
The attenuated ray transform for connections and higgs fields, Geom. Funct. Anal. (GAFA), 22 (2012), 1460-1489.
|
[19] |
G. Paternain, M. Salo and G. Uhlmann,
Tensor tomography on surfaces, Inventiones Math., 193 (2013), 229-247.
|
[20] |
G. P. Paternain, M. Salo and G. Uhlmann,
Spectral rigidity and invariant distributions on anosov surfaces, Journal of Differential Geometry, 98 (2014), 147-181.
doi: 10.4310/jdg/1406137697. |
[21] |
G. P. Paternain, M. Salo and G. Uhlmann,
Invariant distributions, Beurling transforms and tensor tomography in higher dimensions, Mathematische Annalen, 363 (2015), 305-362.
doi: 10.1007/s00208-015-1169-0. |
[22] |
G. P. Paternain and H. Zhou,
Invariant distributions and the geodesic ray transform, Analysis & PDE, 9 (2016), 1903-1930.
doi: 10.2140/apde.2016.9.1903. |
[23] |
L. Pestov and G. Uhlmann,
On the characterization of the range and inversion formulas for the geodesic X-ray transform, International Math. Research Notices, 80 (2004), 4331-4347.
|
[24] |
L. Pestov and G. Uhlmann,
Two-dimensional compact simple Riemannian manifolds are boundary distance rigid, Annals of Mathematics, 161 (2005), 1093-1110.
doi: 10.4007/annals.2005.161.1093. |
[25] |
K. Sadiq, O. Scherzer and A. Tamasan,
On the x-ray transform of planar symmetric 2-tensors, Journal of Mathematical Analysis and Applications, 442 (2016), 31-49.
doi: 10.1016/j.jmaa.2016.04.018. |
[26] |
K. Sadiq and A. Tamasan,
On the range characterization of the two-dimensional attenuated doppler transform, SIAM Journal on Mathematical Analysis, 47 (2015), 2001-2021.
doi: 10.1137/140984282. |
[27] |
K. Sadiq and A. Tamasan,
On the range of the attenuated radon transform in strictly convex sets, Transactions of the American Mathematical Society, 367 (2015), 5375-5398.
|
[28] |
M. Salo and G. Uhlmann,
The Attenuated Ray Transform on Simple Surfaces, J. Diff. Geom., 88 (2011), 161-187.
doi: 10.4310/jdg/1317758872. |
[29] |
V. Sharafutdinov,
Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994. |
[30] |
G. Sparr, K. Strahlen, K. Lindstrom and H. Persson,
Doppler tomography for vector fields, Inverse Problems, 11 (1995), 1051-1061.
doi: 10.1088/0266-5611/11/5/009. |
[31] |
P. Stefanov and G. Uhlmann,
An inverse source problem in optical molecular imaging, Analysis & PDE, 1 (2008), 115-126.
doi: 10.2140/apde.2008.1.115. |
[32] |
E. M. Stein and R. Shakarchi,
Real Analysis. Measure Theory, Integration and Hilbert Spaces, Princeton University Press, 2005. |
[33] |
W. A. Strauss,
Partial Differential Equations, Second edition. John Wiley & Sons, Ltd., Chichester, 2008. |
[34] |
A. Tamasan,
Tomographic reconstruction of vector fields in variable background media, Inverse Problems, 23 (2007), 2197-2205.
doi: 10.1088/0266-5611/23/5/022. |
[35] |
H. Zhou,
Generic injectivity and stability of inverse problems for connections, Communications in Partial Differential Equations, 42 (2017), 780-801.
doi: 10.1080/03605302.2017.1295061. |
show all references
References:
[1] |
G. Ainsworth,
The attenuated magnetic ray transform on surfaces, Inverse Problems and Imaging, 7 (2013), 27-46.
|
[2] |
E. V. Arbuzov, A. L. Bukhgeim and S. G. Kazantsev,
Two-dimensional tomography problems and the theory of A-analytic functions, Siberian Advances in Mathematics, 8 (1998), 1-20.
|
[3] |
Y. M. Assylbekov, F. Monard and G. Uhlmann, Inversion formulas and range characterizations for the attenuated geodesic ray transform,
Journal de Mathématiques Pures et Appliquées (to appear), arXiv: 1609.04361. |
[4] |
G. Bal,
On the attenuated radon transform with full and partial measurements, Inverse Problems, 20 (2004), 399-418.
|
[5] |
G. Bal and A. Tamasan,
Inverse source problem in transport equations, SIAM J. Math. Anal., 39 (2007), 57-76.
|
[6] |
J. Boman and J.-O. Strömberg,
Novikov's inversion formula for the attenuated radon transform--a new approach, J. Geom. Anal., 14 (2004), 185-198.
|
[7] |
D. Finch, The attenuated X-ray transform: Recent developments, Inside Out, Inverse Problems and Applications, 47-66, Math. Sci. Res. Inst. Publ., 47, Cambridge Univ. Press, Cambridge, 2003. |
[8] |
S. Holman and P. Stefanov,
The weighted doppler transform, Inverse Problems and Imaging, 4 (2010), 111-130.
|
[9] |
S. G. Kazantsev and A. A. Bukhgeim,
Inversion of the scalar and vector attenuated x-ray transforms in a unit disc, J. Inv. Ill-Posed Problems, 15 (2007), 735-765.
|
[10] |
S. G. Kazantsev and A. A. Bukhgeim,
Singular value decomposition for the 2d fan-beam radon transform of tensor fields, Journal of Inverse and Ill-posed Problems, 12 (2004), 245-278.
|
[11] |
F. Monard,
Efficient tensor tomography in fan-beam coordinates, Inverse Probl. Imaging, 10 (2016), 433-459.
|
[12] |
F. Monard,
Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces, SIAM J. Math. Anal., 48 (2016), 1155-1177.
|
[13] |
F. Monard and G. P. Paternain, The geodesic X-ray transform with a $GL(n, \mathbb{C})$
-connection,
to appear in Journal of Geometric Analysis, arXiv: 1610.09571. |
[14] |
F. Natterer,
Inversion of the attenuated radon transform, Inverse Problems, 17 (2001), 113-119.
|
[15] |
F. Natterer,
The Mathematics of Computerized Tomography, SIAM, 2001. |
[16] |
R. G. Novikov, An inversion formula for the attenuated x-ray transformation, Ark. Math.,
40 (2002), 145-167, (Rapport de recherche 00/05-3 Université de Nantes, Laboratoire de
Mathématiques). |
[17] |
G. Paternain,
Inside Out II, chapter Inverse problems for connections, 2012. |
[18] |
G. Paternain, M. Salo and G. Uhlmann,
The attenuated ray transform for connections and higgs fields, Geom. Funct. Anal. (GAFA), 22 (2012), 1460-1489.
|
[19] |
G. Paternain, M. Salo and G. Uhlmann,
Tensor tomography on surfaces, Inventiones Math., 193 (2013), 229-247.
|
[20] |
G. P. Paternain, M. Salo and G. Uhlmann,
Spectral rigidity and invariant distributions on anosov surfaces, Journal of Differential Geometry, 98 (2014), 147-181.
doi: 10.4310/jdg/1406137697. |
[21] |
G. P. Paternain, M. Salo and G. Uhlmann,
Invariant distributions, Beurling transforms and tensor tomography in higher dimensions, Mathematische Annalen, 363 (2015), 305-362.
doi: 10.1007/s00208-015-1169-0. |
[22] |
G. P. Paternain and H. Zhou,
Invariant distributions and the geodesic ray transform, Analysis & PDE, 9 (2016), 1903-1930.
doi: 10.2140/apde.2016.9.1903. |
[23] |
L. Pestov and G. Uhlmann,
On the characterization of the range and inversion formulas for the geodesic X-ray transform, International Math. Research Notices, 80 (2004), 4331-4347.
|
[24] |
L. Pestov and G. Uhlmann,
Two-dimensional compact simple Riemannian manifolds are boundary distance rigid, Annals of Mathematics, 161 (2005), 1093-1110.
doi: 10.4007/annals.2005.161.1093. |
[25] |
K. Sadiq, O. Scherzer and A. Tamasan,
On the x-ray transform of planar symmetric 2-tensors, Journal of Mathematical Analysis and Applications, 442 (2016), 31-49.
doi: 10.1016/j.jmaa.2016.04.018. |
[26] |
K. Sadiq and A. Tamasan,
On the range characterization of the two-dimensional attenuated doppler transform, SIAM Journal on Mathematical Analysis, 47 (2015), 2001-2021.
doi: 10.1137/140984282. |
[27] |
K. Sadiq and A. Tamasan,
On the range of the attenuated radon transform in strictly convex sets, Transactions of the American Mathematical Society, 367 (2015), 5375-5398.
|
[28] |
M. Salo and G. Uhlmann,
The Attenuated Ray Transform on Simple Surfaces, J. Diff. Geom., 88 (2011), 161-187.
doi: 10.4310/jdg/1317758872. |
[29] |
V. Sharafutdinov,
Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994. |
[30] |
G. Sparr, K. Strahlen, K. Lindstrom and H. Persson,
Doppler tomography for vector fields, Inverse Problems, 11 (1995), 1051-1061.
doi: 10.1088/0266-5611/11/5/009. |
[31] |
P. Stefanov and G. Uhlmann,
An inverse source problem in optical molecular imaging, Analysis & PDE, 1 (2008), 115-126.
doi: 10.2140/apde.2008.1.115. |
[32] |
E. M. Stein and R. Shakarchi,
Real Analysis. Measure Theory, Integration and Hilbert Spaces, Princeton University Press, 2005. |
[33] |
W. A. Strauss,
Partial Differential Equations, Second edition. John Wiley & Sons, Ltd., Chichester, 2008. |
[34] |
A. Tamasan,
Tomographic reconstruction of vector fields in variable background media, Inverse Problems, 23 (2007), 2197-2205.
doi: 10.1088/0266-5611/23/5/022. |
[35] |
H. Zhou,
Generic injectivity and stability of inverse problems for connections, Communications in Partial Differential Equations, 42 (2017), 780-801.
doi: 10.1080/03605302.2017.1295061. |

[1] |
François Monard. Efficient tensor tomography in fan-beam coordinates. Inverse Problems and Imaging, 2016, 10 (2) : 433-459. doi: 10.3934/ipi.2016007 |
[2] |
Dan Jane, Gabriel P. Paternain. On the injectivity of the X-ray transform for Anosov thermostats. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 471-487. doi: 10.3934/dcds.2009.24.471 |
[3] |
Aleksander Denisiuk. On range condition of the tensor x-ray transform in $ \mathbb R^n $. Inverse Problems and Imaging, 2020, 14 (3) : 423-435. doi: 10.3934/ipi.2020020 |
[4] |
Gareth Ainsworth. The attenuated magnetic ray transform on surfaces. Inverse Problems and Imaging, 2013, 7 (1) : 27-46. doi: 10.3934/ipi.2013.7.27 |
[5] |
François Rouvière. X-ray transform on Damek-Ricci spaces. Inverse Problems and Imaging, 2010, 4 (4) : 713-720. doi: 10.3934/ipi.2010.4.713 |
[6] |
Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems and Imaging, 2021, 15 (5) : 893-928. doi: 10.3934/ipi.2021021 |
[7] |
Hiroshi Fujiwara, Kamran Sadiq, Alexandru Tamasan. Partial inversion of the 2D attenuated $ X $-ray transform with data on an arc. Inverse Problems and Imaging, 2022, 16 (1) : 215-228. doi: 10.3934/ipi.2021047 |
[8] |
Sean Holman, Plamen Stefanov. The weighted Doppler transform. Inverse Problems and Imaging, 2010, 4 (1) : 111-130. doi: 10.3934/ipi.2010.4.111 |
[9] |
Gareth Ainsworth, Yernat M. Assylbekov. On the range of the attenuated magnetic ray transform for connections and Higgs fields. Inverse Problems and Imaging, 2015, 9 (2) : 317-335. doi: 10.3934/ipi.2015.9.317 |
[10] |
Jan Boman. Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform. Inverse Problems and Imaging, 2010, 4 (4) : 619-630. doi: 10.3934/ipi.2010.4.619 |
[11] |
Venkateswaran P. Krishnan, Plamen Stefanov. A support theorem for the geodesic ray transform of symmetric tensor fields. Inverse Problems and Imaging, 2009, 3 (3) : 453-464. doi: 10.3934/ipi.2009.3.453 |
[12] |
Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems and Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023 |
[13] |
Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems and Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649 |
[14] |
Gareth Ainsworth. The magnetic ray transform on Anosov surfaces. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1801-1816. doi: 10.3934/dcds.2015.35.1801 |
[15] |
Venkateswaran P. Krishnan, Vladimir A. Sharafutdinov. Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2021076 |
[16] |
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 |
[17] |
Yiran Wang. Parametrices for the light ray transform on Minkowski spacetime. Inverse Problems and Imaging, 2018, 12 (1) : 229-237. doi: 10.3934/ipi.2018009 |
[18] |
Michael Krause, Jan Marcel Hausherr, Walter Krenkel. Computing the fibre orientation from Radon data using local Radon transform. Inverse Problems and Imaging, 2011, 5 (4) : 879-891. doi: 10.3934/ipi.2011.5.879 |
[19] |
Sunghwan Moon. Inversion of the spherical Radon transform on spheres through the origin using the regular Radon transform. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1029-1039. doi: 10.3934/cpaa.2016.15.1029 |
[20] |
Victor Palamodov. Remarks on the general Funk transform and thermoacoustic tomography. Inverse Problems and Imaging, 2010, 4 (4) : 693-702. doi: 10.3934/ipi.2010.4.693 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]