Реферат: The practical achievements of emission tomography, especially in medical diagnostics and biology, are well known. In the emission tomography problem it is assumed that the object under consideration contains internal sources the radiation of which is fixed by detectors. The problem is to nd the source distribution. Here we propose the concept generalized attenuated ray transforms as generalization within the framework of the model for emission tomography. The integral operators are the heart of tomography approaches, consisting in non-destructive mode of obtaining information, which is accumulated along lines or curves of integration. A list of integral transforms, describing initial data for integral geometry and tomography problems, is very vast [1]. Weighted longitudinal ray transforms of tensor elds [2], [3] stand out among the others as they are the solutions of differential equations of transport type. We prove the uniqueness theorems of boundary-value and initial boundary-value problems for the derived equations. Thus, the integral geometry and tensor tomography problems can be treated as inverse problems of right-hand part reconstruction by known certain families of weighted ray transforms of tensor fields. Next variety of integral operators applies for investigations of integral geometry and refraction tensor tomography problems. A generalization of back-projection operator provides the operators of integral angular moments for weighted ray transforms of tensor fields and functions f(x; \xi) depending on a point x and direction vector \xi. Differential operators of tensor analysis, plying to tensor elds in conjunction with the operators of weighted ray transforms and integral angular moments, present favorable tools for investigation of integral geometry and tomography problems [2]. Some part of them can be considered as conservation laws in framework of iterative methods for approximate solutions of refraction tensor tomography problems. The work has been supported by the Program of fundamental researches of SB RAS No. I.1.5 (project 0314-2016-0011), RFBR and DFG according to the research project No. 19-51-12008. REFERENCES 1. Sharafutdinov V.A. Integral geometry of tensor elds // Utrecht: VSP, The Netherlands, 1994. 2. Derevtsov E.Yu., Svetov I.E. Tomography of tensor elds in the plane // Eurasian J. Math. Comp. Applications, V. 3 (2), 2015, pp. 24-68. 3. Derevtsov E.Yu. On a generalization of exponential ray transform in tomography // J. Math. Sciences, 2021, Vol. 253, No. 3, pp. 369-381.

Библиографическая ссылка: Derevtsov E.Y.
Attenuated ray transforms and angular moments in integral geometry and tomography
В сборнике The XIII international scientific conference and young scientist school "Theory and Numerics of Inverse and Ill-posed Problems" Novosibirsk, Akademgorodok, April 12-22, 2021. 2021.

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