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Journal of Applied and Industrial Mathematics

, Volume 9, Issue 4, pp 447–460 | Cite as

Reconstruction of the singular support of a tensor field given in a refracting medium by its ray transform

  • E. Yu. Derevtsov
  • S. V. Maltseva
Article

Abstract

We propose some approaches for numerically solving the problem of reconstructing the singular support of a symmetric tensor field, given in a refracting medium, by its known ray transform. To solve the problem, we use the back-projection operators that act on the ray transforms and the tensor analysis methods on Riemannian manifolds. We construct the operators of the medium inhomogeneity indicator that allow us to identify the sets of points of the singular support of the scalar, vector, and tensor fields. We propose and implement algorithms for solving the problem under study.

Keywords

tomography tensor field function discontinuity singular support refraction ray transform back-projection operator tensor analysis 

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References

  1. 1.
    E. I. Vainberg, I. A. Kazak, and M. L. Faingoiz, “X-Ray Computerized Back Projection Tomography with Filtration by Double Differentiation. Procedure and Information Features,” Soviet J. Nondest. Test. No. 21, pp.106–113 (1985).Google Scholar
  2. 2.
    A. Faridani, E. L. Ritman, and K. T. Smith, “Local Tomography,” SIAM J. Appl. Math. 52 (2), 459–484 (1992).MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    A. Faridani, D. V. Finch, E. L. Ritman, and K. T. Smith, “Local Tomography. II,” SIAM J. Appl.Math. 57 (4), 1095–1127 (1997).MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    A. K. Louis and P. Maass, “Contour Reconstruction in 3-D X-Ray CT,” Trans.Med. Imag. 12 (4), 764–769 (1993).CrossRefGoogle Scholar
  5. 5.
    D. S. Anikonov, “Application of Peculiarities of the Transport Equation Solution in the X-Ray Tomography,” Dokl. Ross. Akad. Nauk 335 (6), 702–704 (1994) [Phys. Dokl. 39 (4), 205–207 (1994)].MathSciNetGoogle Scholar
  6. 6.
    D. S. Anikonov, “A Special Problem of Integral Geometry,” Dokl. Ross. Akad Nauk 415 (1), 7–9 (2007). [Dokl.Math. 76 (1), 483–485 (2007)].MathSciNetGoogle Scholar
  7. 7.
    E. Yu. Derevtsov, “Some Approaches to the Reconstruction of a Singular Support of Scalar, Vector, and Tensor Fields by Their Known Tomographic Data,” Sibirsk. Elektron.Mat. Izv. 5, 632–646 (2008).MATHMathSciNetGoogle Scholar
  8. 8.
    E. Yu. Derevtsov and V. V. Pikalov, “Reconstruction of Vector Fields and Their Singularities by Ray Transforms,” Sibirsk. Zh. Vychisl.Mat. 14 (1), 29–46 (2011) [Numer. Anal. Appl. 4 (1), 21–35 (2011)].MATHGoogle Scholar
  9. 9.
    V. Sharafutdinov, M. Skokan, and G. Uhlmann, “Regularity of Ghosts in Tensor Tomography,” J. Geom. Anal. 15 (3), 499–542 (2005).MathSciNetCrossRefGoogle Scholar
  10. 10.
    V. A. Sharafutdinov, Integral Geometry of Tensor Fields (Nauka, Novosibirsk, 1993; VSP, Utrecht, 1994).MATHGoogle Scholar
  11. 11.
    N. E. Kochin, Vector Calculus and Basics of Tensor Calculus (ONTI, Leningrad, 1934) [in Russian].Google Scholar
  12. 12.
    H. Weyl, “TheMethod ofOrthogonal Projection in Potential Theory,” DukeMath. J. No. 7, 411–444 (1940).MathSciNetCrossRefGoogle Scholar
  13. 13.
    I. S. Gradsteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Fizmatgiz, Moscow, 1963; Academic Press, New York, 1980).Google Scholar
  14. 14.
    F. Monard, “On Reconstruction Formulas for the Ray Transform Acting on Symmetric Differentials on Surfaces,” Inverse Problems (2014) 30 (6); URL: arXiv:1311.6167v2[math.AP]Google Scholar
  15. 15.
    B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry.Methods and Applications (Nauka, Moscow, 1986; Springer, New York, 1992).MATHGoogle Scholar
  16. 16.
    N. S. Bakhvalov, Numerical Methods (Nauka, Moscow, 1975; Mir, Moscow, 1977).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia
  3. 3.Lavrent’ev Institute of HydrodynamicsNovosibirskRussia

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