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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

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