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Differential Equations and Uniqueness Theorems for the Generalized Attenuated Ray Transforms of Tensor Fields

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Book cover Numerical Computations: Theory and Algorithms (NUMTA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11974))

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Abstract

Properties of operators of generalized attenuated ray transforms (ART) are investigated. Starting with Radon transform in the mathematical model of computer tomography, attenuated ray transform in emission tomography and longitudinal ray transform in tensor tomography, we come to the operators of ART of order k over symmetric m-tensor fields, depending on spatial and temporal variables. The operators of ART of order k over tensor fields contain complex-valued absorption, different weights, and depend on time. Connections between ART of various orders are established by means of application of linear part of transport equation. This connections lead to the inhomogeneous k-th order differential equations for the ART of order k over symmetric m-tensor field. The right hand parts of such equations are m-homogeneous polynomials containing the components of the tensor field as the coefficients. The polynomial variables are the components \(\xi ^j\) of direction vector \(\xi \) participating in differential part of transport equation. Uniqueness theorems of boundary-value and initial boundary-value problems for the obtained equations are proved, with significant application of Gauss-Ostrogradsky theorem. The connections of specified operators with integral geometry of tensor fields, emission tomography, photometry and wave optics allow to treat the problem of inversion of the ART of order k as the inverse problem of determining the right hand part of certain differential equation.

The reported study was funded by Russian Foundation for Basic Research (RFBR) and German Science Foundation (DFG) according to the joint German-Russian research project 19-51-12008 and by German Science Foundation (DFG) under project Lo 310/17-1.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics SB RAS, 630090, Novosibirsk, Russia

    Evgeny Yu. Derevtsov & Yuriy S. Volkov

  2. Novosibirsk State University, 630090, Novosibirsk, Russia

    Evgeny Yu. Derevtsov & Yuriy S. Volkov

  3. Saarland University, 66041, Saarbrücken, Germany

    Thomas Schuster

Authors
  1. Evgeny Yu. Derevtsov
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  2. Yuriy S. Volkov
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  3. Thomas Schuster
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Correspondence to Yuriy S. Volkov .

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Editors and Affiliations

  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

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Derevtsov, E.Y., Volkov, Y.S., Schuster, T. (2020). Differential Equations and Uniqueness Theorems for the Generalized Attenuated Ray Transforms of Tensor Fields. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11974. Springer, Cham. https://doi.org/10.1007/978-3-030-40616-5_8

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  • DOI: https://doi.org/10.1007/978-3-030-40616-5_8

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  • Online ISBN: 978-3-030-40616-5

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