Skip to main content
SpringerLink
Log in

Reconstruction of a Vector Field in a Ball from its Normal Radon Transform

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We study a vector tomography problem of reconstructing potential components of a three-dimensional vector field from its normal Radon transform. The method of singular value decomposition is used. For the subspace of potential fields with potentials vanishing on the boundary, we construct an orthogonal basis and compute its image under the normal Radon transform.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. F. Natterer, The Mathematics of Computerized Tomography, JohnWiley & Sons etc., Chichester etc. (1986).

  2. A. K. Louis, “Orthogonal function series expansions and the null space of the Radon transform,” SIAM J. Math. Anal. 15, No. 3, 621–633 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  3. E. Yu. Derevtsov, A. V. Efimov, A. K. Louis, and T. Schuster, “Singular value decomposition and its application to numerical inversion for ray transforms in 2d vector tomography,” J. Inverse Ill-Posed Probl. 19, No. 4–5, 689–715 (2011).

    MATH  MathSciNet  Google Scholar 

  4. E. Yu. Derevtsov and A. P. Polyakova, “Solution of the integral geometry problem for 2-tensor fields by the singular value decomposition method” [in Russian], Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform. 12, No. 3, 73–94 (2012); English transl.: J. Math. Sci., New York 202, No. 1, 50–71 (2014).

  5. S. G. Kazantsev and A. A. Bukhgeim, “Singular value decomposition for the 2d fan-beam Radon transform of tensor fields,” J. Inverse Ill-Posed Probl. 12, No. 3, 245–278 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  6. H. Weyl, “The method of orthogonal projection in potential theory,” Duke Math. J. No. 7, 411–444 (1940).

    Article  MathSciNet  Google Scholar 

  7. S. Deans, The Radon Transform and Some of Its Applications, John Wiley & Sons etc., New YOrk etc. (1983).

  8. E. Yu. Derevtsov, Tomography of Composed Media. Models, Methods, and Algorithms. I: Model of Scalar Tomography [in Russian], Gorno-Altai. Gos. Univ. Press, Gorno-Altaisk (2009).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics SB RAS, 4, pr. Akad. Koptyuga, Novosibirsk, 630090, Russia

    A. P. Polyakova

Authors
  1. A. P. Polyakova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. P. Polyakova.

Additional information

Translated from Vestnik Novosibirskogo Gosudarstvennogo Universiteta: Seriya Matematika, Mekhanika, Informatika 13, No. 4, 2013, pp. 119-142.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Polyakova, A.P. Reconstruction of a Vector Field in a Ball from its Normal Radon Transform. J Math Sci 205, 418–439 (2015). https://doi.org/10.1007/s10958-015-2256-1

Download citation

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-015-2256-1

Keywords

Search

Navigation