Skip to main content
SpringerLink
Log in

Approximate reconstruction from ray integrals of a function on a domain with low refraction

  • Published:
Journal of Applied and Industrial Mathematics Aims and scope Submit manuscript

Abstract

An approach is proposed for reconstruction from ray integrals of a function defined on a Riemannian domain with low refraction. Using the back-projection operator and the fast Fourier transform, an inversion algorithm is constructed for the ray transform and its numerical study is carried out.

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. J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Ber. Verh. Sächs. Akad. Wiss. Leipzig Math. Nat. Kl. 69, 262–277 (1917).

    Google Scholar 

  2. S. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

    MATH  Google Scholar 

  3. F. Natterer, The Mathematics of Computerized Tomography (Teubner-Wiley, Stutgart, 1986; Mir, Moscow, 1990).

    MATH  Google Scholar 

  4. E. Yu. Derevtsov, A. G. Kleshchev, and V. A. Sharafutdinov, “Numerical Solution of the Emission 2D-Tomography Problem for a Medium with Absorption and Refraction,” J. Inverse Ill-Posed Probl. 7(1), 83–103 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  5. E. Yu. Derevtsov, I. E. Svetov, and Yu. S. Volkov, “The B-Spline Application to the Problem of Emission 2D-Tomography in a Medium with Refraction,” Sibirsk. Zh. Industr. Mat. 11(3), 45–60 (2008).

    MATH  MathSciNet  Google Scholar 

  6. I. E. Svetov, E. Yu. Derevtsov, Yu. S. Volkov, and T. Schuster, “A Numerical Solver Based on B-Splines for 2D Vector Field Tomography in a Refracting Medium,” Math. Comput. Simulation. 97, 207–223 (2014).

    Article  MathSciNet  Google Scholar 

  7. A. K. Louis, “Approximate Inverse for Linear and Some Nonlinear Problems,” Inverse Probl. 12, 175–190 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  8. E.Yu. Derevtsov, R. Dietz, A.K. Louis, and T. Schuster, “Influence of Refraction to the Accuracy of a Solution for the 2D-Emission Tomography Problem,” J. Inverse Ill-Posed Probl. 8(2), 161–191 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  9. V. V. Pikalov, “Reconstruction of the Tomogram of a Transparent Inhomogeneity by the Inverted Wave Method,” Optika i Spectroskopiya 65(4), 956–962 (1988).

    Google Scholar 

  10. A. K. Louis, “Eikonal Approximation in Ultrasound Computerized Tomography Signal Processing. II,” in Control and Applications (Springer, New York, 1990), pp. 285–291.

    Google Scholar 

  11. T. Pfitzenreiter and T. Schuster, “Tomographic Reconstruction of the Curl and Divergence of 2DVector Fields Taking Refractions into Account,” SIAM J. Imaging Sci. 4, 40–56 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  12. V. G. Romanov, “The Representation of Functions by Integrals on a Family of Curves,” Sibirsk. Mat. Zh. 8(5), 1206–1208 (1967) [Siberian Math. J. 8 (5), 923–925 (1967)].

    MATH  MathSciNet  Google Scholar 

  13. A. M. Cormack, “The Radon Transform on a Family of Curves in the Plane. I,” Proc. AMS. 83(2), 325–330 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  14. A. M. Cormack, “The Radon Transform on a Family of Curves in the Plane. II,” Proc. AMS. 86(2), 293–298 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  15. Statistical Methods for Digital Computers, Ed. by K. Enslein, E. Ralston, and G. S. Wilf (Wiley, New York, 1977; Nauka, Moscow, 1986).

    MATH  Google Scholar 

  16. K. Jano and S. Bochner, Curvature and Betti Numbers (Princeton Univ., Princeton, NJ, 1953; Inostrannaya Literatura, Moscow, 1957).

    Google Scholar 

  17. G. N. Lance, Numerical Methods for High Speed Computers (Iliffe & Sons, London, 1960; Inostrannaya Literatura, Moscow, 1962).

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    E. Yu. Derevtsov, S.V. Maltseva & I. E. Svetov

  2. Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

    E. Yu. Derevtsov, S.V. Maltseva & I. E. Svetov

Authors
  1. E. Yu. Derevtsov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S.V. Maltseva
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. I. E. Svetov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Yu. Derevtsov.

Additional information

Original Russian Text © E.Yu. Derevtsov, S.V. Maltseva, I.E. Svetov, 2014, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2014, Vol. XVII, No. 4, pp. 48–59.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Derevtsov, E.Y., Maltseva, S. & Svetov, I.E. Approximate reconstruction from ray integrals of a function on a domain with low refraction. J. Appl. Ind. Math. 9, 36–46 (2015). https://doi.org/10.1134/S1990478915010056

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1990478915010056

Keywords

Search

Navigation