Реферат: Integral and differential operators are important tools at settings, investigations and solving methods for problems of integral geometry, tensor and refraction tomography. Integral transforms represent the essence of tomography approaches, consisting in nondestructive mode of obtaining information, which is accumulated along lines of integration. At present a list of integral operators, describing initial data, is very vast. It includes such operators as generalized Radon transforms, weighted longitudinal, transverse and mixed ray transforms of tensor fields [1]-[3]. Second family of integral operators is applied for investigations and solving the problems of integral geometry and refractive tensor tomography. It should be recalled that inversion formulas for the Radon transform contain integral operators of back-projection, Riesz potential, Fourier and Hilbert transforms. A generalization of back-projection operator provides the operators of angular moments for attenuated ray transforms of tensor fields, et al. Differential operators of tensor analysis and the operators of angular moments, present useful tools for investigation of integral geometry and tomography problems [3]. Properties of attenuated weighted ray transforms and angular moments of ray transforms for tensor fields are established. The differential equations for generalized ray transforms and their connections are investigated. Acknowledgements. The reported study was funded partially by RFBR and DFG according to the research project 19-51-12008. References [1] Svetov I. E., Derevtsov E.Yu., Volkov Yu. S., Schuster T. (2014) A numerical solver based on B-splines for 2D vector field tomography in a refracting medium. Mathematics and Computers in Simulation, Vol. 97, pp. 207-223. [2] Derevtsov E.Yu., Svetov I. E. (2015) Tomography of tensor elds in the plane. Eurasian J. Math. Comp. Applications, Vol. 3, no. 2, pp. 24-68. [3] Derevtsov E.Yu. (2018) On a generalization of attenuated ray transform in tomography. Sib. J. of Pure and Applied Math., Vol. 18, no. 4, pp. 29-41.

Библиографическая ссылка: Volkov Y.S. , Derevtsov E.Y. , Шустер Т.
Integral and differential operators as the tools of integral geometry and tomography
В сборнике Book of Abstracts of the 3rd International Conference and Summer School Numerical Computations: Theory and Algorithms. 2019. – C.243. – ISBN 9788874581016.

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