TY - JOUR

T1 - Generalized attenuated ray transforms and their integral angular moments

AU - Derevtsov, Evgeny Yu

AU - Volkov, Yuriy S.

AU - Schuster, Thomas

N1 - Publisher Copyright: © 2020 Copyright: Copyright 2020 Elsevier B.V., All rights reserved. Publisher Copyright: © 2020

PY - 2021/11/15

Y1 - 2021/11/15

N2 - In this article generalized attenuated ray transforms (ART) and integral angular moments are investigated. Starting from the Radon transform, the attenuated ray transform and the longitudinal ray transform, we derive the concept of ART-operators of order k over functions defined on the phase space and depending on time. The ART-operators are generalized for complex-valued absorption coefficient as well as weight functions of polynomial and exponential type. Connections between ART operators of various orders are established by means of the application of the linear part of a transport equation. These connections lead to inhomogeneous differential equations of order (k+1) for the ART of order k. Uniqueness theorems for the corresponding boundary-value and initial boundary-value problems are proved. Properties of integral angular moments of order p are considered and connections between the moments of different orders are deduced. A close connection of the considered operators with mathematical models for tomography, physical optics and integral geometry allows to treat the inversion of ART of order k as an inverse problem of determining the right-hand side of a corresponding differential equation.

AB - In this article generalized attenuated ray transforms (ART) and integral angular moments are investigated. Starting from the Radon transform, the attenuated ray transform and the longitudinal ray transform, we derive the concept of ART-operators of order k over functions defined on the phase space and depending on time. The ART-operators are generalized for complex-valued absorption coefficient as well as weight functions of polynomial and exponential type. Connections between ART operators of various orders are established by means of the application of the linear part of a transport equation. These connections lead to inhomogeneous differential equations of order (k+1) for the ART of order k. Uniqueness theorems for the corresponding boundary-value and initial boundary-value problems are proved. Properties of integral angular moments of order p are considered and connections between the moments of different orders are deduced. A close connection of the considered operators with mathematical models for tomography, physical optics and integral geometry allows to treat the inversion of ART of order k as an inverse problem of determining the right-hand side of a corresponding differential equation.

KW - Attenuated ray transform

KW - Boundary-value problem

KW - Integral angular moment

KW - Tomography

KW - Transport equation

KW - Uniqueness theorem

UR - http://www.scopus.com/inward/record.url?scp=85087659634&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2020.125494

DO - 10.1016/j.amc.2020.125494

M3 - Article

AN - SCOPUS:85087659634

VL - 409

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

M1 - 125494

ER -