Skip to main content
SpringerLink
Log in

The Method of Approximate Inverse in Slice-by-Slice Vector Tomography Problems

  • Conference paper
  • First Online:
Book cover Numerical Computations: Theory and Algorithms (NUMTA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11974))

Abstract

A numerical solution of the problem of recovering the solenoidal part of three-dimensional vector field using the incomplete tomographic data is proposed. Namely, values of the ray transform for all straight lines, which are parallel to one of the coordinate planes, are known. The recovery algorithms are based on the method of approximate inverse.

This research was partially supported by RFBR and DFG according to the research project 19-51-12008.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Sharafutdinov, V.A.: Integral Geometry of Tensor Fields. VSP, Utrecht (1994). https://doi.org/10.1515/9783110900095

  2. Denisjuk, A.: Inversion of the x-ray transform for 3D symmetric tensor fields with sources on a curve. Inverse Prob. 22(2), 399–411 (2006). https://doi.org/10.1088/0266-5611/22/2/001

    Article  MathSciNet  MATH  Google Scholar 

  3. Sharafutdinov, V.: Slice-by-slice reconstruction algorithm for vector tomography with incomplete data. Inverse Prob. 23(6), 2603–2627 (2007). https://doi.org/10.1088/0266-5611/23/6/021

    Article  MathSciNet  MATH  Google Scholar 

  4. Svetov, I.E.: Reconstruction of the solenoidal part of a three-dimensional vector field by its ray transforms along straight lines parallel to coordinate planes. Numer. Anal. Appl. 5(3), 271–283 (2012). https://doi.org/10.1134/S1995423912030093

    Article  MATH  Google Scholar 

  5. Svetov, I.: Slice-by-slice numerical solution of \(3D\)-vector tomography problem. In: Kosmas, T., Vagenas, E., Vlachos, D. (eds.) Journal of Physics: Conference Series, IC-MSQUARE 2012, vol. 410, p. 012042. IOP Publishing Ltd., Bristol (2013). https://doi.org/10.1088/1742-6596/410/1/012042

    Google Scholar 

  6. Derevtsov, E.Y., Efimov, A.V., Louis, A.K., Schuster, T.: Singular value decomposition and its application to numerical inversion for ray transforms in 2D vector tomography. J. Inverse Ill-posed Probl. 19(4–5), 689–715 (2011). https://doi.org/10.1515/jiip.2011.047

    Article  MathSciNet  MATH  Google Scholar 

  7. Svetov, I.E., Derevtsov, E.Y., Volkov, Y.S., Schuster, T.: A numerical solver based on B-splines for 2D vector field tomography in a refracting medium. Math. Comput. Simul. 97, 207–223 (2014). https://doi.org/10.1016/j.matcom.2013.10.002

    Article  MathSciNet  Google Scholar 

  8. Derevtsov, E.Y., Svetov, I.E.: Tomography of tensor fields in the plain. Eurasian J. Math. Comput. Appl. 3(2), 24–68 (2015)

    Google Scholar 

  9. Polyakova, A.: Reconstruction of potential part of 3D vector field by using singular value decomposition. In: Kosmas, T., Vagenas, E., Vlachos, D. (eds.) IC-MSQUARE 2012, Journal of Physics: Conference Series, vol. 410, p. 012015. IOP Publishing Ltd., Bristol (2013). https://doi.org/10.1088/1742-6596/410/1/012015

    Google Scholar 

  10. Polyakova, A.P.: Reconstruction of a vector field in a ball from its normal Radon transform. J. Math. Sci. 205(3), 418–439 (2015). https://doi.org/10.1007/s10958-015-2256-1

    Article  MathSciNet  Google Scholar 

  11. Polyakova, A.P., Svetov, I.E.: Numerical solution of the problem of reconstructing a potential vector field in the unit ball from its normal Radon transform. J. Appl. Indu. Math. 9(4), 547–558 (2015). https://doi.org/10.1134/S1990478915040110

    Article  MATH  Google Scholar 

  12. Louis, A.K.: Inverse und schlecht gestellte Probleme. Vieweg+Teubner Verlag, Stuttgart (1989).https://doi.org/10.1007/978-3-322-84808-6

    Book  Google Scholar 

  13. Louis, A.K., Maass, P.: A mollifier method for linear operator equations of the first kind. Inverse Prob. 6(3), 427–440 (1990). https://doi.org/10.1088/0266-5611/6/3/011

    Article  MathSciNet  MATH  Google Scholar 

  14. Louis, A.K.: Approximate inverse for linear and some nonlinear problems. Inverse Prob. 12(2), 175–190 (1996). https://doi.org/10.1088/0266-5611/12/2/005

    Article  MATH  Google Scholar 

  15. Rieder, A., Schuster, T.: The approximate inverse in action with an application to computerized tomography. SIAM J. Numer. Anal. 37(6), 1909–1929 (2000). https://doi.org/10.1137/S0036142998347619

    Article  MathSciNet  MATH  Google Scholar 

  16. Rieder, A., Schuster, T.: The approximate inverse in action III: 3D-Doppler tomography. Numer. Math. 97(2), 353–378 (2004). https://doi.org/10.1007/s00211-003-0512-7

    Article  MathSciNet  MATH  Google Scholar 

  17. Schuster, T.: Defect correction in vector field tomography: detecting the potential part of a field using BEM and implementation of the method. Inverse Prob. 21(1), 75–91 (2005). https://doi.org/10.1088/0266-5611/21/1/006

    Article  MathSciNet  MATH  Google Scholar 

  18. Svetov, I., Maltseva, S., Polyakova, A.: Numerical solution of 2D-vector tomography problem using the method of approximate inverse. In: Ashyralyev, A., Lukashov, A. (eds.), ICAAM 2016, AIP Conference Proceedings, vol. 1759, p. 020132. AIP Publishing LLC, Melville (2016). https://doi.org/10.1063/1.4959746

  19. Derevtsov, E.Y., Louis, A.K., Maltseva, S.V., Polyakova, A.P., Svetov, I.E.: Numerical solvers based on the method of approximate inverse for 2D vector and 2-tensor tomography problems. Inverse Prob. 33(12), 124001 (2017). https://doi.org/10.1088/1361-6420/aa8f5a

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Ivan E. Svetov & Svetlana V. Maltseva

  2. Saarland University, Saarbrucken, Germany

    Alfred K. Louis

Authors
  1. Ivan E. Svetov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Svetlana V. Maltseva
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alfred K. Louis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivan E. Svetov .

Editor information

Editors and Affiliations

  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Svetov, I.E., Maltseva, S.V., Louis, A.K. (2020). The Method of Approximate Inverse in Slice-by-Slice Vector Tomography Problems. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11974. Springer, Cham. https://doi.org/10.1007/978-3-030-40616-5_47

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-40616-5_47

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-40615-8

  • Online ISBN: 978-3-030-40616-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation