American Institute of Mathematical Sciences

Advanced
June  2019, 13(3): 679-701. doi: 10.3934/ipi.2019031

Momentum ray transforms

Venkateswaran P. Krishnan 1, , Ramesh Manna 1, , Suman Kumar Sahoo 1, and  Vladimir A. Sharafutdinov 2,3,

1. 

TIFR Centre for Applicable Mathematics, Sharada Nagar, Chikkabommasandra, Yelahanka New Town, Bangalore, India
2. 

Sobolev Institute of Mathematics; 4 Koptyug Avenue, Novosibirsk, 630090, Russia
3. 

Novosibirsk State University, 2 Pirogov street, 630090, Russia

Received  July 2018 Revised  October 2018 Published  March 2019

Fund Project: The first author was supported by US NSF grant DMS 1616564 and a SERB Matrics Grant, MTR/2017/000837.
The second author was supported by SERB National Postdoctoral fellowship, PDF/2017/002780.
The first three authors were supported by Airbus Corporate Foundation Chair grant "Mathematics of Complex Systems" established at TIFR CAM and TIFR ICTS, Bangalore, India.
The work was started when the last author visited TIFR CAM January 2017. The author is grateful to the institute for the support and hospitality.
The last author was supported by RFBR, Grant 17-51-150001.

The momentum ray transform $ I^k $ integrates a rank $ m $ symmetric tensor field $ f $ over lines in $ \mathbb{R}^n $ with the weight $ t^k $: $ (I^k\!f)(x,\xi) = \int_{-\infty}^\infty t^k\langle f(x+t\xi),\xi^m\rangle\, \mathrm{d} t. $ In particular, the ray transform $ I = I^0 $ was studied by several authors since it had many tomographic applications. We present an algorithm for recovering $ f $ from the data $ (I^0\!f,I^1\!f,\dots, I^m\!f) $. In the cases of $ m = 1 $ and $ m = 2 $, we derive the Reshetnyak formula that expresses $ \|f\|_{H^s_t({\mathbb R}^n)} $ through some norm of $ (I^0\!f,I^1\!f,\dots, I^m\!f) $. The $ H^{s}_{t} $-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.

Keywords: Ray transform, Reshetnyak formula, inverse problems, tensor analysis, stability estimates.
Mathematics Subject Classification: Primary: 44A12, 65R32; Secondary: 46F12.
Citation: Venkateswaran P. Krishnan, Ramesh Manna, Suman Kumar Sahoo, Vladimir A. Sharafutdinov. Momentum ray transforms. Inverse Problems and Imaging, 2019, 13 (3) : 679-701. doi: 10.3934/ipi.2019031
References:
[1]

A. Abhishek and R. K. Mishra, Support theorems and an injectivity result for integral moments of a symmetric m-tensor field, https://arXiv.org/abs/1704.02010, Journal of Fourier Analysis and Applications, 2018, 1–26. doi: 10.1007/s00041-018-09649-7.

[2]

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

[3]

W. Lionheart and V. A. Sharafutdinov, Reconstruction algorithm for the linearized polarization tomography problem with incomplete data, in Imaging Microstructures: Mathematical and Computational Challenges, Ed. Habib Ammari and Hyeonbae Kang, Contemporary Mathematics, 494 (2009), 137–159. doi: 10.1090/conm/494/09648.

[4]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-posed Problems Series. VSP, Utrecht, 1994. doi: 10.1515/9783110900095.

[5]

V. A. Sharafutdinov, The Reshetnyak formula and Natterer stability estimates in tensor tomography, Inverse Problems, 33 (2017), 025002, 20pp. doi: 10.1088/1361-6420/33/2/025002.

[6]

V. A. Sharafutdinov and J.-N. Wang, Tomography of small residual stresses, Inverse Problems, 28 (2012), 065017, 17 pp. doi: 10.1088/0266-5611/28/6/065017.

show all references

References:
[1]

A. Abhishek and R. K. Mishra, Support theorems and an injectivity result for integral moments of a symmetric m-tensor field, https://arXiv.org/abs/1704.02010, Journal of Fourier Analysis and Applications, 2018, 1–26. doi: 10.1007/s00041-018-09649-7.

[2]

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

[3]

W. Lionheart and V. A. Sharafutdinov, Reconstruction algorithm for the linearized polarization tomography problem with incomplete data, in Imaging Microstructures: Mathematical and Computational Challenges, Ed. Habib Ammari and Hyeonbae Kang, Contemporary Mathematics, 494 (2009), 137–159. doi: 10.1090/conm/494/09648.

[4]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-posed Problems Series. VSP, Utrecht, 1994. doi: 10.1515/9783110900095.

[5]

V. A. Sharafutdinov, The Reshetnyak formula and Natterer stability estimates in tensor tomography, Inverse Problems, 33 (2017), 025002, 20pp. doi: 10.1088/1361-6420/33/2/025002.

[6]

V. A. Sharafutdinov and J.-N. Wang, Tomography of small residual stresses, Inverse Problems, 28 (2012), 065017, 17 pp. doi: 10.1088/0266-5611/28/6/065017.

[1]

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
[2]

Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas and Michael Taylor. Metric tensor estimates, geometric convergence, and inverse boundary problems. Electronic Research Announcements, 2003, 9: 69-79.

[3]

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
[4]

Jan Boman, Vladimir Sharafutdinov. Stability estimates in tensor tomography. Inverse Problems and Imaging, 2018, 12 (5) : 1245-1262. doi: 10.3934/ipi.2018052
[5]

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
[6]

Mehdi Badra, Fabien Caubet, Jérémi Dardé. Stability estimates for Navier-Stokes equations and application to inverse problems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2379-2407. doi: 10.3934/dcdsb.2016052
[7]

Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems and Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010
[8]

Guillaume Bal, Alexandre Jollivet. Stability estimates in stationary inverse transport. Inverse Problems and Imaging, 2008, 2 (4) : 427-454. doi: 10.3934/ipi.2008.2.427
[9]

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
[10]

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
[11]

Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems and Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551
[12]

Guillaume Bal, Alexandre Jollivet. Generalized stability estimates in inverse transport theory. Inverse Problems and Imaging, 2018, 12 (1) : 59-90. doi: 10.3934/ipi.2018003
[13]

Pu-Zhao Kow, Jenn-Nan Wang. Refined instability estimates for some inverse problems. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022017
[14]

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
[15]

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
[16]

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
[17]

Xinchi Huang, Masahiro Yamamoto. Carleman estimates for a magnetohydrodynamics system and application to inverse source problems. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022005
[18]

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
[19]

Siamak RabieniaHaratbar. Support theorem for the Light-Ray transform of vector fields on Minkowski spaces. Inverse Problems and Imaging, 2018, 12 (2) : 293-314. doi: 10.3934/ipi.2018013
[20]

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

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (229)
  • HTML views (276)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy