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

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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 (formula presented) of direction vector (formula presented) 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.

Original languageEnglish
Title of host publicationNumerical Computations
Subtitle of host publicationTheory and Algorithms - 3rd International Conference, NUMTA 2019, Revised Selected Papers
EditorsYaroslav D. Sergeyev, Dmitri E. Kvasov, Yaroslav D. Sergeyev, Dmitri E. Kvasov
PublisherSpringer Gabler
Pages97-111
Number of pages15
ISBN (Print)9783030406158
DOIs
Publication statusPublished - 1 Jan 2020
Event3rd Triennial International Conference and Summer School on Numerical Computations: Theory and Algorithms, NUMTA 2019 - Crotone, Italy
Duration: 15 Jun 201921 Jun 2019

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11974 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference3rd Triennial International Conference and Summer School on Numerical Computations: Theory and Algorithms, NUMTA 2019
CountryItaly
CityCrotone
Period15.06.201921.06.2019

Keywords

  • Attenuated ray transform
  • Boundary-value problem
  • Tensor tomography
  • Transport equation
  • Uniqueness theorem
  • DYNAMIC INVERSE PROBLEMS
  • OPTICAL COHERENCE TOMOGRAPHY
  • EFFICIENT ALGORITHMS
  • VECTOR-FIELDS
  • REGULARIZATION
  • POLARIZATION

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