Skip to main content
SpringerLink
Log in

On Phase Correction in Tomographic Research

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

Abstract

Under consideration is the problem of improving the contrast of the image obtained by processing tomographic projections with phase distortion. The study is based on the well-known intensity transfer equation. Unlike other works, this equation is solved in a bounded region of variation of the tomographic parameters. In a domain, a boundary value problem is posed for the intensity transfer equation which is then specialized for a three-dimensional parallel tomographic scheme. The case of two-dimensional tomography is also considered, together with the corresponding boundary value problem for the intensity transfer equation. We propose numerical methods for solving the boundary value problems of phase correction. The results are given of the numerical experiments on correction of tomographic projections and reconstruction of the structure of the objects under study (in particular, a slice of a geological sample) by using piecewise uniform regularization.

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

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.

Similar content being viewed by others

REFERENCES

  1. D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous Phase and Amplitude Extraction from a Single Defocused Image of a Homogeneous Object,” J. Microscopy 206, 33–40 (2001).

    Article  MathSciNet  Google Scholar 

  2. S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson and S. W. Wilkins, “X-Ray Phase-Contrast Microscopy and Microtomography,” Optics Express 11 (19), 2289–2302 (2003).

    Article  Google Scholar 

  3. M. A. Beltran, D. M. Paganin, K. Uesugi, and M. J. Kitchen, “2D and 3D X-Ray Phase Retrieval of Multi-Material Objects Using a Single Defocus Distance,” Optics Express 18 (7), 6423–6436 (2010).

    Article  Google Scholar 

  4. A. A. Samarskii, Introduction to the Theory of Difference Schemes (Nauka, Moscow, 1971) [in Russian].

    Google Scholar 

  5. N. N. Kalitkin, Numerical Methods (Nauka, Moscow, 1978) [in Russian].

    Google Scholar 

  6. A. N. Tikhonov and V. Ya. Arsenin, Methods of Solving Ill-Posed Problems (Nauka, Moscow, 1979) [in Russian].

    MATH  Google Scholar 

  7. V. K. Ivanov, V. V. Vasin, and V. P. Tanana, Theory of Linear Ill-Posed Problems and Its Applications (Nauka, Moscow, 1978) [in Russian].

    MATH  Google Scholar 

  8. V.A.Morozov, Regular Methods of Solving Ill-Posed Problems (Nauka, Moscow, 1987) [in Russian].

    Google Scholar 

  9. A. N. Tikhonov, A. V. Goncharskii, V. V. Stepanov, and A. G. Yagola, Numerical Methods of Solving Ill-Posed Problems (Nauka, Moscow, 1990) [in Russian].

    Google Scholar 

  10. A. N. Tikhonov, A. S. Leonov, and A. G. Yagola, Nonlinear Ill-Posed Problems (Nauka, Moscow, 1995; Kurs, Moscow, 2017) [in Russian].

  11. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, Dordrecht, 1996).

    Book  Google Scholar 

  12. A. S. Leonov, Solution of Ill-Posed Inverse Problems. An Outline of the Theory, Practical Algorithms, and Demonstrations by MATLAB (Knizhn. Dom “LIBROKOM”, Moscow, 2009) [in Russian].

    Google Scholar 

  13. A. S. Leonov, “Functions of Several Variables of Bounded Variation in the Ill-Posed Problems,” Zh. Vychisl. Mat. Mat. Fiz. 36 (9), 35–49 (1996).

    MathSciNet  Google Scholar 

  14. A. S. Leonov, “Some Remarks on Complete Variation of Functions of Several Variables and a Multidimensional Analog of Helly’s Choice Principle,” Mat. Zametki 63 (1), 69–80 (1998).

    Article  MathSciNet  Google Scholar 

  15. A. S. Leonov, “Numerical Piecewise-Uniform Regularization for Two-Dimensional Ill-Posed Problems,” Inverse Problems 15, 1165–1176 (1999).

    Article  MathSciNet  Google Scholar 

  16. A. S. Leonov, Y. Wang, and A. G. Yagola, “Piecewise Uniform Regularization for the Inverse Problem of Microtomography with A Posteriori Error Estimate,” Inverse Probl. Sci. Engrg. 27, 1–11 (2019).

    Article  Google Scholar 

  17. Y. F. Wang, S. F. Fan, A. S. Leonov, D. V. Lukyanenko, A. G. Yagola, L. H. Wang , Yu Wang, and J. Q. Wang, “Non-Smooth Regularization and Fast Optimization Algorithm for Micropore Reconstruction of a Shale,” Chinese J. Geophysics 63 (5), 2036–2042 (2019).

    Google Scholar 

Download references

Funding

The authors were supported by the Russian Foundation for Basic Research (project no. 19–51–53005–GFEN–a) and the Competitiveness Increase Program of National Research Nuclear University “MEPhI” (project no. 02.a03.21.0005).

Author information

Authors and Affiliations

  1. Key Laboratory of Petroleum Resources Research, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, 100029, China

    Ya. Wang

  2. National Research Nuclear University (MEPhI), Moscow, 115409, Russia

    A. S. Leonov

  3. Faculty of Physics, Lomonosov Moscow State University, Moscow, 119991, Russia

    D. V. Lukyanenko, V. D. Shinkarev & A. G. Yagola

Authors
  1. Ya. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. S. Leonov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. V. Lukyanenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. V. D. Shinkarev
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. A. G. Yagola
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Ya. Wang, A. S. Leonov, D. V. Lukyanenko or A. G. Yagola.

Additional information

Translated by B.L. Vertgeim

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Y., Leonov, A.S., Lukyanenko, D.V. et al. On Phase Correction in Tomographic Research. J. Appl. Ind. Math. 14, 802–810 (2020). https://doi.org/10.1134/S1990478920040171

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1990478920040171

Keywords

Search

Navigation