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The statement and numerical solution of an optimization problem in X-ray tomography

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Abstract

A nonclassical problem is considered for the transport equation with coefficients depending on the energy of radiation. The task is to find the discontinuity surfaces for the coefficients of the equation from measurements of the radiation flux leaving the medium. For this tomography problem, an optimization problem is stated and numerically analyzed. The latter consists in determining the radiation energy that ensures the best reconstruction of the unknown medium. A simplified optimization problem is solved analytically.

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References

  1. V. S. Vladimirov, “Mathematical Problems in One-Speed Theory of Particle Transport,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 61, 3–158 (1961).

    Google Scholar 

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

    Google Scholar 

  3. D. S. Anikonov, A. E. Kovtanyuk, and I. V. Prokhorov, The Use of Transport Equations in Tomography (Logos, Moscow, 2000) [in Russian].

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  6. D. S. Konovalova, “A Method for Approximation of a Visibility Measure in X-Ray Tomography,” Sib. Zh. Ind. Mat. 8(1(21)), 64–69 (2005).

    Google Scholar 

  7. A. I. Akhiezer and V. B. Berestetskii, Quantum Electrodynamics (Nauka, Moscow, 1981) [in Russian].

    Google Scholar 

  8. J. H. Hubbell and S. M. Seltzer, Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients, 1 keV to 20 MeV for Elements Z = 1 to 92 and 48 Additional Substances of Dosimetric Interest,” NISTIR-5632 (Natl. Inst. Standard. Technol., Gaithersburg, 1995).

    Google Scholar 

  9. G. I. Marchuk, G. A. Mikhailov, M. A. Nazarliev, et al., Monte Carlo Method in Atmospheric Optics (Nauka, Novosibirsk, 1976) [in Russian].

    Google Scholar 

  10. S. M. Ermakov and G. A. Mikhailov, Statistical Simulation (Nauka, Moscow, 1982) [in Russian].

    Google Scholar 

  11. F. Natterer, The Mathematics of Computerized Tomography (Wiley, Chichester, 1986; Mir, Moscow, 1990).

    Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Division, Russian Academy of Sciences, pr. Akademika Koptyuga 4, Novosibirsk, 630090, Russia

    D. S. Anikonov

  2. Institute of Applied Mathematics, Far East Division, Russian Academy of Sciences, ul. Radio 7, Vladivostok, 690041, Russia

    I. V. Prokhorov

Authors
  1. D. S. Anikonov
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  2. I. V. Prokhorov
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Original Russian Text © D.S. Anikonov, I.V. Prokhorov, 2006, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2006, Vol. 46, No. 1, pp. 18–25.

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Anikonov, D.S., Prokhorov, I.V. The statement and numerical solution of an optimization problem in X-ray tomography. Comput. Math. and Math. Phys. 46, 16–22 (2006). https://doi.org/10.1134/S0965542506010040

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  • DOI: https://doi.org/10.1134/S0965542506010040

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