Реферат: A problem of recovery of a tensor field and its discontinuities by certain types of ray transforms is investigated. Detailed decomposition of a symmetric tensor field, its representation through potentials and application of ray transforms [1] allow to reconstruct the field by means of suggested in [2] back-projection operators. The description of kernels and images of new types of ray transforms, the inversion formulas for sought-for tensor field or for its potentials are obtained. The problem of reconstruction of discontinuities of a tensor field given in refracting medium is considered. To solve this problem, we construct some operators of the inhomogeneity indicator, which enables to identify the set of points of discontinuities of the field. Back-projection operators and differential operators of tensor analysis are used. The operators form a set of algorithms for reconstruction of discontinuities [3]. A generalization for attenuated ray transform (ART) is suggested [4]. Connections between ART of various orders are established and their differential equations are obtained. Angular moments of ART as generalization of back-projection operators are developed. The work was funded partially by RFBR and DFG according to the research project No. 19-51-12008. References 1. Derevtsov E.Yu., Svetov I. E. Tomography of tensor fields in the plane. Eurasian J. Math. Comp. Appl. 2015. V. 3, N. 2. P. 24–68. 2. Derevtsov E.Yu., Maltseva S. V., Svetov I. E. Mathematical models and algorithms for reconstruction of singular support of functions and vector fields by tomographic data. Eurasian J. Math. Comp. Appl. 2015. V. 3, N. 4. P. 4–44. 3. Derevtsov E.Yu., Maltseva S. V., Svetov I. E. Determination of Discontinuities of a Function in a Domain with Refraction from Its Attenuated Ray Transform. J. Applied and Industrial Mathematics. 2018. V. 12, N. 4. P. 619–641. 4. Derevtsov E.Yu. On a generalization of attenuated ray transform in tomography. Sib. J. Pure Applied Math. 2018. V. 18, N. 4. P. 29–41.

Библиографическая ссылка: Derevtsov E.Y.
A reconstruction of a tensor field and its discontinuities by tomography data,
В сборнике Марчуковские научные чтения - 2020. Тезисы Международной конференции, посвященной 95-летию со дня рождения акад. Г. И. Марчука 19‒23 октября 2020 г. Академгородок, Новосибирск, Россия. 2020. – C.119-120. – ISBN 978-5-4437-1108-9. DOI: 10.24411/9999-017A-2020-10198

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