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Sibirskii Zhurnal Vychislitel'noi Matematiki, 2011, Volume 14, Number 1, Pages 29–46 (Mi sjvm424)  
This article is cited in 15 scientific papers (total in 15 papers)

Reconstruction of vector fields and their singularities from ray transforms

E. Yu. Derevtsovab, V. V. Pickalovc

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b Novosibirsk State University, Novosibirsk
c Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
Full-text PDF (1385 kB) Citations (15)
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Abstract: In this paper, numerical methods for reconstruction of the singular support of a vector field from its known longitudinal and (or) transverse ray transforms are proposed. Apart from a modification for the Vainberg operator, we use integral operators of angular moments and back projections as well as differential operators of tensor analysis for solving the problem. Results of numerical simulation for reconstructing discontinuous vector fields and with discontinuities in the derivatives are presented. Visualization of their singular support is shown.
Key words: vector tomography, tensor field, Radon transform, ray transform, angular moment, back projection, inversion formulas, singular support, visualization.
Received: 01.06.2010
Revised: 15.06.2010
English version:
Numerical Analysis and Applications, 2011, Volume 4, Issue 1, Pages 21–35
DOI: https://doi.org/10.1134/S1995423911010034
Bibliographic databases:
Document Type: Article
UDC: 517.98+519.677
Language: Russian
Citation: E. Yu. Derevtsov, V. V. Pickalov, “Reconstruction of vector fields and their singularities from ray transforms”, Sib. Zh. Vychisl. Mat., 14:1 (2011), 29–46; Num. Anal. Appl., 4:1 (2011), 21–35
Citation in format AMSBIB
\Bibitem{DerPic11}
\by E.~Yu.~Derevtsov, V.~V.~Pickalov
\paper Reconstruction of vector fields and their singularities from ray transforms
\jour Sib. Zh. Vychisl. Mat.
\yr 2011
\vol 14
\issue 1
\pages 29--46
\mathnet{http://mi.mathnet.ru/sjvm424}
\transl
\jour Num. Anal. Appl.
\yr 2011
\vol 4
\issue 1
\pages 21--35
\crossref{https://doi.org/10.1134/S1995423911010034}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79952477299}
Linking options:
  • https://www.mathnet.ru/eng/sjvm424
  • https://www.mathnet.ru/eng/sjvm/v14/i1/p29
    • This publication is cited in the following 15 articles:
    1. Gaik Ambartsoumian, Mohammad J. Latifi Jebelli, Rohit K. Mishra, “Numerical Implementation of Generalized V-Line Transforms on 2D Vector Fields and their Inversions”, SIAM J. Imaging Sci., 17:1 (2024), 595  crossref
    2. David Omogbhe, Kamran Sadiq, “On the X-ray transform of planar symmetric tensors”, Journal of Inverse and Ill-posed Problems, 2023  crossref
    3. Derevtsov E.Yu., Maltseva V S., “Recovery of a Vector Field in the Cylinder By Its Jointly Known Nmr Images and Ray Transforms”, Sib. Electron. Math. Rep., 18 (2021), 86–103  mathnet  crossref  mathscinet  isi  scopus
    4. V V Pickalov, “Approximate filtered back-projection algorithm for plane curves in tomography problems”, J. Phys.: Conf. Ser., 1715:1 (2021), 012039  crossref
    5. Ambartsoumian G., Jebelli Mohammad Javad Latifi, Mishra R.K., “Generalized V-Line Transforms in 2D Vector Tomography”, Inverse Probl., 36:10 (2020), 104002  crossref  mathscinet  zmath  isi  scopus
    6. Krishnan V.P., Mishra R.K., Monard F., “On Solenoidal-Injective and Injective Ray Transforms of Tensor Fields on Surfaces”, J. Inverse Ill-Posed Probl., 27:4 (2019), 527–538  crossref  mathscinet  zmath  isi  scopus
    7. E. Yu. Derevtsov, S. V. Maltseva, I. E. Svetov, “Determination of discontinuities of a function in a domain with refraction from its attenuated ray transform”, J. Appl. Industr. Math., 12:4 (2018), 619–641  mathnet  crossref  crossref  elib  elib
    8. Kazantsev D., Pickalov V., Nagella S., Pasca E., Withers Ph.J., “Tomophantom, a Software Package to Generate 2D-4D Analytical Phantoms For Ct Image Reconstruction Algorithm Benchmarks”, SoftwareX, 7 (2018), 150–155  crossref  isi  scopus
    9. A. Koulouri, M. Brookes, V. Rimpilaeinen, “Vector tomography for reconstructing electric fields with non-zero divergence in bounded domains”, J. Comput. Phys., 329 (2017), 73–90  crossref  mathscinet  isi  scopus
    10. E. Yu. Derevtsov, S. V. Maltseva, I. E. Svetov, M. A. Sultanov, “On a problem of reconstruction of discontinuities of a function by its attenuated ray transform along geodesics”, International Conference Functional Analysis in Interdisciplinary Applications (FAIA 2017), AIP Conf. Proc., 1880, eds. T. Kalmenov, M. Sadybekov, Amer. Inst. Phys., 2017, UNSP 060004  crossref  isi  scopus
    11. F. Monard, “Efficient tensor tomography in fan-beam coordinates”, Inverse Probl. Imaging, 10:2 (2016), 433–459  crossref  isi
    12. D. V. Kurlikovskii, A. L. Ageev, T. V. Antonova, “Issledovanie porogovogo (korrelyatsionnogo) metoda i ego prilozhenie k lokalizatsii osobennostei”, Sib. elektron. matem. izv., 13 (2016), 829–848  mathnet  crossref  isi
    13. E. Yu. Derevtsov, S. V. Maltseva, I. E. Svetov, M. A. Sultanov, “On a problem of reconstruction of a discontinuous function by its Radon transform”, International Conference on Analysis and Applied Mathematics (ICAAM 2016), AIP Conf. Proc., 1759, eds. A. Ashyralyev, A. Lukashov, Amer. Inst. Phys., 2016, 020019  crossref  isi
    14. E. Yu. Derevtsov, S. V. Maltseva, “Reconstruction of a singular support of a tensor field given in refractive medium by its ray transform”, J. Appl. Industr. Math., 9:4 (2015), 447–460  mathnet  crossref  crossref  mathscinet  elib
    15. T. V. Antonova, “Localization method for lines of discontinuity of approximately defined function of two variables”, Num. Anal. Appl., 5:4 (2012), 285–296  mathnet  crossref  elib
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