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 Sibirsk. Mat. Zh., 2012, Volume 53, Number 4, Pages 721–740 (Mi smj2359)

A polychromatic inhomogeneity indicator in an unknown medium for an $X$-ray tomography problem

D. S. Anikonova, E. Yu. Balakinab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b Novosibirsk State University, Novosibirsk

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.

Keywords: unknown boundary, transport equation, inverse problem, tomography.

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English version:
Siberian Mathematical Journal, 2012, 53:4, 573–590

Bibliographic databases:

UDC: 517.958

Citation: D. S. Anikonov, E. Yu. Balakina, “A polychromatic inhomogeneity indicator in an unknown medium for an $X$-ray tomography problem”, Sibirsk. Mat. Zh., 53:4 (2012), 721–740; Siberian Math. J., 53:4 (2012), 573–590

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. E. Yu. Balakina, “Numerical realization of the algorithm of reconstruction of an inhomogeneous medium for an X-ray tomography problem”, J. Appl. Industr. Math., 8:2 (2014), 158–167
2. E. Yu. Balakina, “Layerwise sensing in $X$-ray tomography in the polychromatic case”, Comput. Math. Math. Phys., 54:2 (2014), 335–352
3. V. G. Romanov, “Recovering jumps in X-ray tomography”, J. Appl. Industr. Math., 8:4 (2014), 582–593
4. Romanov V.G., “Reconstruction of Discontinuities in a Problem of Integral Geometry”, Dokl. Math., 90:3 (2014), 758–761
5. E. Yu. Balakina, “Suschestvovanie i edinstvennost resheniya dlya nestatsionarnogo uravneniya perenosa”, Sib. zhurn. industr. matem., 18:4 (2015), 3–8
6. E. Yu. Balakina, “Finding discontinuities in the coefficients of the linear nonstationary transport equations”, Comput. Math. Math. Phys., 57:10 (2017), 1650–1665
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