Momentum ray transforms

Venkateswaran P. Krishnan; Ramesh Manna; Suman Kumar Sahoo; Vladimir A. Sharafutdinov

doi:10.3934/ipi.2019031

Article Contents

2019, Volume 13, Issue 3: 679-701. Doi: 10.3934/ipi.2019031

This issue Previous Article A dual EM algorithm for TV regularized Gaussian mixture model in image segmentation Next Article

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: June 30, 2018

Revised: September 30, 2018

Published: March 2019

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 44A12, 65R32; Secondary: 46F12.

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References

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