Abstract
We pose and study an X-ray tomography problem, which is an inverse problem for the transport differential equation, making account for particle absorption by a medium and single scattering. The statement of the problem corresponds to a stage-by-stage probing of the unknown medium common in practice. Another step towards a more realistic problem is the use of integrals over energy of the density of emanating radiation flux as the known data, in contrast to specifying the flux density for every energy level, as it is customary in tomography. The required objects are the discontinuity surfaces of the coefficients of the equation, which corresponds to searching for the boundaries between various substances contained in the medium. We prove a uniqueness theorem for the solution under quite general assumptions and a condition ensuring the existence of the required surfaces. The proof is rather constructive in character and suitable for creating a numerical algorithm.
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Original Russian Text Copyright © 2012 Anikonov D.S. and Balakina E.Yu.
The authors were supported by the Russian Foundation for Basic Research (Grants 10-01-00384-a and 11-08-00286-a) and the Federal Target Program “Scientific and Educational Personnel of Innovation Russia” for 2009-2013 (State Contract 16.740.11.0127).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 53, No. 4, pp. 721–740, July–August, 2012.
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Anikonov, D.S., Balakina, E.Y. A polychromatic inhomogeneity indicator in an unknown medium for an X-ray tomography problem. Sib Math J 53, 573–590 (2012). https://doi.org/10.1134/S0037446612040015
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DOI: https://doi.org/10.1134/S0037446612040015