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Momentum ray transforms. / Krishnan, Venkateswaran P.; Manna, Ramesh; Sahoo, Suman Kumar et al.

In: Inverse Problems and Imaging, Vol. 13, No. 3, 06.2019, p. 679-701.

Research output: Contribution to journal › Article › peer-review

Harvard

Krishnan, VP, Manna, R, Sahoo, SK & Sharafutdinov, VA 2019, 'Momentum ray transforms', Inverse Problems and Imaging, vol. 13, no. 3, pp. 679-701. https://doi.org/10.3934/ipi.2019031

APA

Krishnan, V. P., Manna, R., Sahoo, S. K., & Sharafutdinov, V. A. (2019). Momentum ray transforms. Inverse Problems and Imaging, 13(3), 679-701. https://doi.org/10.3934/ipi.2019031

Vancouver

Krishnan VP, Manna R, Sahoo SK, Sharafutdinov VA. Momentum ray transforms. Inverse Problems and Imaging. 2019 Jun;13(3):679-701. doi: 10.3934/ipi.2019031

Author

Krishnan, Venkateswaran P. ; Manna, Ramesh ; Sahoo, Suman Kumar et al. / Momentum ray transforms. In: Inverse Problems and Imaging. 2019 ; Vol. 13, No. 3. pp. 679-701.

BibTeX

@article{2d9a3f9246b1485daf6c8aa2eba39bad,
title = "Momentum ray transforms",
abstract = " The momentum ray transform I k integrates a rank m symmetric tensor field f over lines in R n with the weight t k : (I k f)(x,ξ)=∫ ∞ -∞ t k 〈 f(x+tξ),ξ m〉 dt. In particular, the ray transform I=I0 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,…,I m f). In the cases of m=1 and m=2, we derive the Reshetnyak formula that expresses ∥f∥Hst(ℝn) through some norm of (I 0 f,I 1 f,…,I m f). The Hst-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 = "Inverse problems, Ray transform, Reshetnyak formula, Stability estimates, Tensor analysis, tensor analysis, inverse problems, stability estimates",
author = "Krishnan, {Venkateswaran P.} and Ramesh Manna and Sahoo, {Suman Kumar} and Sharafutdinov, {Vladimir A.}",
note = "Publisher Copyright: {\textcopyright} 2019 American Institute of Mathematical Sciences.",
year = "2019",
month = jun,
doi = "10.3934/ipi.2019031",
language = "English",
volume = "13",
pages = "679--701",
journal = "Inverse Problems and Imaging",
issn = "1930-8337",
publisher = "American Institute of Mathematical Sciences",
number = "3",

}

RIS

TY - JOUR

T1 - Momentum ray transforms

AU - Krishnan, Venkateswaran P.

AU - Manna, Ramesh

AU - Sahoo, Suman Kumar

AU - Sharafutdinov, Vladimir A.

N1 - Publisher Copyright: © 2019 American Institute of Mathematical Sciences.

PY - 2019/6

Y1 - 2019/6

N2 - The momentum ray transform I k integrates a rank m symmetric tensor field f over lines in R n with the weight t k : (I k f)(x,ξ)=∫ ∞ -∞ t k 〈 f(x+tξ),ξ m〉 dt. In particular, the ray transform I=I0 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,…,I m f). In the cases of m=1 and m=2, we derive the Reshetnyak formula that expresses ∥f∥Hst(ℝn) through some norm of (I 0 f,I 1 f,…,I m f). The Hst-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.

AB - The momentum ray transform I k integrates a rank m symmetric tensor field f over lines in R n with the weight t k : (I k f)(x,ξ)=∫ ∞ -∞ t k 〈 f(x+tξ),ξ m〉 dt. In particular, the ray transform I=I0 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,…,I m f). In the cases of m=1 and m=2, we derive the Reshetnyak formula that expresses ∥f∥Hst(ℝn) through some norm of (I 0 f,I 1 f,…,I m f). The Hst-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.

KW - Inverse problems

KW - Ray transform

KW - Reshetnyak formula

KW - Stability estimates

KW - Tensor analysis

KW - tensor analysis

KW - inverse problems

KW - stability estimates

UR - http://www.scopus.com/inward/record.url?scp=85065743163&partnerID=8YFLogxK

U2 - 10.3934/ipi.2019031

DO - 10.3934/ipi.2019031

M3 - Article

AN - SCOPUS:85065743163

VL - 13

SP - 679

EP - 701

JO - Inverse Problems and Imaging

JF - Inverse Problems and Imaging

SN - 1930-8337

IS - 3

ER -

ID: 20157883