Skip to main content
SpringerLink
Log in

Determination of the Attenuation Coefficient for the Nonstationary Radiative Transfer Equation

  • MATHEMATICAL PHYSICS
  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

For the nonstationary radiative transfer equation, the inverse problem of determining the attenuation coefficient from a known solution at the domain boundary is considered. The structure and the continuous properties of the solution to an initial-boundary value problem for the radiative transfer equation are studied. Under special assumptions about the radiation source, it is shown that the inverse problem has a unique solution and a formula for the Radon transform of the attenuation coefficient is derived. The quality of the reconstructed tomographic images of the sought function is analyzed numerically in the case of various angular and time flux density distributions of the external source.

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.

Similar content being viewed by others

REFERENCES

  1. G. I. Marchuk, “Formulations of some inverse problems,” Dokl. Akad. Nauk SSSR 156 (3), 503–506 (1964).

    MathSciNet  Google Scholar 

  2. M. V. Maslennikov, “The Milne problem with anisotropic scattering,” Proc. Steklov Inst. Math. 97, 1–161 (1968).

    MathSciNet  MATH  Google Scholar 

  3. D. S. Anikonov, “On inverse problems for the transport equation,” Differ. Uravn. 2 (1), 7–17 (1974).

    MathSciNet  Google Scholar 

  4. A. I. Prilepko and A. L. Ivankov, “Inverse problems of determining a coefficient, scattering phase function, and right side of a nonstationary multispeed transport equation,” Differ. Uravn. 21 (5), 870–885 (1985).

    MATH  Google Scholar 

  5. D. S. Anikonov, “Uniqueness of determining a coefficient of a transfer equation with a special-type source,” Dokl. Akad. Nauk SSSR 284 (5), 1033–1037 (1985).

    MathSciNet  Google Scholar 

  6. D. G. Orlovskii and A. I. Prilepko, “Some inverse problems for the linearized Boltzmann equation,” USSR Comput. Math. Math. Phys. 27 (6), 58–65 (1987).

    Article  MathSciNet  Google Scholar 

  7. D. S. Anikonov and I. V. Prokhorov, “Determination of the coefficient of a transfer equation with energy and angular singularities of external radiation,” Dokl. Akad. Nauk 327 (2), 205–207 (1992).

    Google Scholar 

  8. D. S. Anikonov, I. V. Prokhorov, and A. E. Kovtanyuk, “Investigation of scattering and absorbing media by the methods of X-ray tomography,” J. Inverse Ill-Posed Probl. 1 (4), 259–282 (1993).

    MathSciNet  MATH  Google Scholar 

  9. V. S. Antyufeev and A. N. Bondarenko, “X-ray tomography in scattering media,” SIAM J. Appl. Math. 56 (2), 573–587 (1996).

    Article  MathSciNet  Google Scholar 

  10. V. G. Romanov, “A stability estimate in the problem of determining the dispersion index and relaxation for the transport equation,” Sib. Math. J. 37 (2), 308–324 (1996).

    Article  Google Scholar 

  11. D. S. Anikonov, A. E. Kovtanyuk, and I. V. Prokhorov, Transport Equation and Tomography (VSP, Boston, 2002).

    MATH  Google Scholar 

  12. D. S. Anikonov, V. G. Nazarov, and I. V. Prokhorov, Poorly Visible Media in X-Ray Tomography (VSP, Boston, 2002).

    MATH  Google Scholar 

  13. S. A. Tereshchenko, Computed Tomography Methods (Fizmatlit, Moscow, 2004) [in Russian].

    MATH  Google Scholar 

  14. A. E. Kovtanyuk and I. V. Prokhorov, “Numerical solution of the inverse problem for the polarized-radiation transfer equation,” Numer. Anal. Appl. 1 (1), 46–57 (2008).

    Article  Google Scholar 

  15. I. V. Prokhorov, I. P. Yarovenko, and V. G. Nazarov, “Optical tomography problems at layered media,” Inverse Probl. 24 (2), 025019 (2008).

    Article  MathSciNet  Google Scholar 

  16. N. P. Volkov, “Solvability of certain inverse problems for the nonstationary kinetic transport equation,” Comput. Math. Math. Phys. 56 (9), 1598–1603 (2016).

    Article  MathSciNet  Google Scholar 

  17. G. Bal, “Inverse Transport Theory and Applications,” Inverse Probl. 25 (5), 053001 (2009).

    Article  MathSciNet  Google Scholar 

  18. M. Bellassoued and Y. Boughanja, “An inverse problem for the linear Boltzmann equation with a time-dependent coefficient,” Inverse Probl. 35 (8), 085003 (2019).

    Article  MathSciNet  Google Scholar 

  19. I. K. Chen and D. Kawagoe, “Propagation of boundary-induced discontinuity in stationary radiative transfer and its application to the optical tomography,” Inverse Probl. Imaging 13 (2), 337–351 (2019).

    Article  MathSciNet  Google Scholar 

  20. D. Kawagoe and I.-K. Chen, “Propagation of boundary-induced discontinuity in stationary radiative transfer,” J. Stat. Phys. 170 (1), 127–140 (2018).

    Article  MathSciNet  Google Scholar 

  21. C. Steiding, D. Kolditz, and W. A. Kalender, “A quality assurance framework for the fully automated and objective evaluation of image quality in cone-beam computed tomography,” Medic. Phys. 41, 031901 (2014).

    Article  Google Scholar 

  22. V. S. Kuznetsov, O. V. Nikolaeva, L. P. Bass, A. V. Bykov, and A. V. Priezzhev, “Modeling of ultrashort light pulse propagation in strongly scattering medium,” Mat. Model. 21 (4), 3–14 (2009).

    MathSciNet  MATH  Google Scholar 

  23. G. V. Fetisov, “X-ray diffraction methods for structural diagnostics of materials: progress and achievements,” Phys.-Usp. 63 (1), 2–32 (2020).

    Article  Google Scholar 

  24. Yu. I. Ershov and S. B. Shikhov, Mathematical Fundamentals of Transfer Theory (Energoatomizdat, Moscow, 1985) [in Russian].

    Google Scholar 

  25. V. S. Vladimirov, “Mathematical problems of the uniform-speed theory of transport,” Tr. Mat. Inst. im. V.A. Steklov Akad. Nauk SSSR 61, 3–158 (1961).

    Google Scholar 

  26. C. Cercignani, Theory and Application of the Boltzmann Equation (Elsevier, New York, 1975).

    MATH  Google Scholar 

  27. V. M. Novikov and S. B. Shikhov, Theory of Parametric Effect on Neutron Transport (Energoizdat, Moscow, 1982) [in Russian].

    Google Scholar 

  28. T. A. Germogenova, Local Properties of Solutions to Transport Equations (Nauka, Moscow, 1986) [in Russian].

    MATH  Google Scholar 

  29. V. I. Agoshkov, Boundary Value Problems for Transport Equations (Birkhäuser, Boston, MA, 1998).

    Book  Google Scholar 

  30. F. Natterer, The Mathematics of Computerized Tomography (Wiley, Chichester 1986).

    Book  Google Scholar 

  31. G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, et al., The Monte Carlo Methods in Atmospheric Optics (Nauka, Novosibirsk, 1976; Springer-Verlag, Berlin, 1980).

  32. G. A. Mikhailov and I. N. Medvedev, Optimization of Weighted Algorithms for Statistical Modeling (Omega Print, Novosibirsk, 2011) [in Russian].

    Google Scholar 

  33. W. A. Kalender, Computed Tomography: Fundamentals, System Technology, Image Quality, Applications, 3rd ed. (Publicis, Erlangen, 2011).

    MATH  Google Scholar 

  34. P. Mah, T. E. Reeves, and W. D. McDavid, “Deriving Hounsfield units using grey levels in cone beam computed tomography,” Dentomaxillofac Radiol. 39, 323–335 (2010).

    Article  Google Scholar 

  35. R. Pauwels, R. Jacobs, S. R. Singer, and M. Mupparapu, “CBCT-Based bone quality assessment: Are Hounsfield units applicable?” Dentomaxillofac Radiol. 44 (1), 20140238 (2015).

    Article  Google Scholar 

Download references

Funding

This work was supported by the Russian Foundation for Basic Research (project no. 20-01-00173) and by the Ministry of Science and Higher Education of the Russian Federation (agreement nos. 075-01095-20-00, 075-02-2020-1482-1).

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences, 690041, Vladivostok, Russia

    I. V. Prokhorov & I. P. Yarovenko

  2. Far Eastern Federal University, 690950, Vladivostok, Russia

    I. V. Prokhorov

Authors
  1. I. V. Prokhorov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. P. Yarovenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to I. V. Prokhorov or I. P. Yarovenko.

Additional information

Translated by I. Ruzanova

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Prokhorov, I.V., Yarovenko, I.P. Determination of the Attenuation Coefficient for the Nonstationary Radiative Transfer Equation. Comput. Math. and Math. Phys. 61, 2088–2101 (2021). https://doi.org/10.1134/S0965542521120101

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542521120101

Keywords:

Search

Navigation