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

Momentum ray transforms

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.

Citation: 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
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]

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.

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

Charles A. Stuart. Stability analysis for a family of degenerate semilinear parabolic problems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5297-5337. doi: 10.3934/dcds.2018234

[20]

Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems & Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139

2017 Impact Factor: 1.465

Metrics

  • PDF downloads (35)
  • HTML views (153)
  • Cited by (0)

[Back to Top]