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

Derevtsov, E. Yu.; Volkov, Yu. S.; Schuster, Thomas

doi:10.1007/978-3-030-40616-5_8

Cited by 3 publications

(4 citation statements)

References 29 publications

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“…Especially in the Euclidean geometry (n = 1) condition (3.21) is valid for any positive α 0 . This is in accordance with the results in [9].…”

Section: Simple Calculations Showsupporting

confidence: 93%

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Well-defined forward operators in dynamic diffractive tensor tomography using viscosity solutions of transport equations

Lukas¹,

Schuster²

2021

Preprint

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We consider a general setting for dynamic tensor field tomography in an inhomogeneous refracting and absorbing medium as inverse source problem for the associated transport equation. Following Fermat's principle the Riemannian metric in the considered domain is generated by the refractive index of the medium. There is wealth of results for the inverse problem of recovering a tensor field from its longitudinal ray transform in a static euclidean setting, whereas there are only few inversion formulas and algorithms existing for general Riemannian metrics and time-dependent tensor fields. It is a well-known fact that tensor field tomography is equivalent to an inverse source problem for a transport equation where the ray transform serves as given boundary data. We prove that this result extends to the dynamic case. Interpreting dynamic tensor tomography as inverse source problem represents a holistic approach in this field. To guarantee that the forward mappings are well-defined, it is necessary to prove existence and uniqueness for the underlying transport equations. Unfortunately, the bilinear forms of the associated weak formulations do not satisfy the coercivity condition. To this end we transfer to viscosity solutions and prove their unique existence in appropriate Sobolev (static case) and Sobolev-Bochner (dynamic case) spaces under a certain assumption that allows only small variations of the refractive index. Numerical evidence is given that the viscosity solution solves the original transport equation if the viscosity term turns to zero.Keywords attenuated refractive dynamic ray transform of tensor fields • geodesics • transport equation • viscosity solutions TFT has many possible applications. One is the reconstruction of velocity fields of liquids and gases. This can be used, e.g., to represent blood flows in medicine. TFT is also used in electron tomography, industry, geo-and astrophysics to name only a few application fields. Pioneered by Norton [20] in 1988, fundamental results on Doppler tomography

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“…Especially in the Euclidean geometry (n = 1) condition (3.21) is valid for any positive α 0 . This is in accordance with the results in [9].…”

Section: Simple Calculations Showsupporting

confidence: 93%

“…This is an extension of Sharafutdinov's result in [36] to time-dependent fields with absorption. For constant refractive index n a similar result is found in [9].…”

Section: Fig 1: Sketch Of a Geodesic Curve With Parametrizationsupporting

confidence: 81%

Well-defined forward operators in dynamic diffractive tensor tomography using viscosity solutions of transport equations

Lukas¹,

Schuster²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Especially in the Euclidean geometry (n = 1), the condition (3.18) is valid for any positive α 0 . This is in accordance with the results in [9].…”

supporting

confidence: 93%

“…This is an extension of Sharafutdinov's result in [42] to time-dependent fields with absorption. For constant refractive index n, a similar result is found in [9].…”

Section: Etna Kent State University and Johann Radon Institute (Ricam)supporting

confidence: 80%

Well-defined forward operators in dynamic diffractive tensor tomography using viscosity solutions of transport equations

Lukas¹,

Schuster²

2022

etna

Self Cite

View full text Add to dashboard Cite

We consider a general setting for dynamic tensor field tomography in an inhomogeneous refracting and absorbing medium as an inverse source problem for the associated transport equation. Following Fermat's principle, the Riemannian metric in the considered domain is generated by the refractive index of the medium. There is a wealth of results for the inverse problem of recovering a tensor field from its longitudinal ray transform in a static Euclidean setting, whereas there are only a few inversion formulas and algorithms existing for general Riemannian metrics and time-dependent tensor fields. It is a well-known fact that tensor field tomography is equivalent to an inverse source problem for a transport equation where the ray transform serves as given boundary data. We prove that this result extends to the dynamic case. Interpreting dynamic tensor tomography as an inverse source problem represents a holistic approach in this field. To guarantee that the forward mappings are well defined, it is necessary to prove existence and uniqueness for the underlying transport equations. Unfortunately, the bilinear forms of the associated weak formulations do not satisfy the coercivity condition. To this end we transfer to viscosity solutions and prove their unique existence in appropriate Sobolev (static case) and Sobolev-Bochner (dynamic case) spaces under a certain assumption that allows only small variations of the refractive index. Numerical evidence is given that the viscosity solution solves the original transport equation if the viscosity term turns to zero.

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Inversion problem for Radon transforms defined on pseudoconvex sets

Anikonov,

Konovalova

2024

Doklady Rossijskoj akademii nauk. Matematika, informatika, proc

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This paper is devoted to some questions of inversion for the classical and generalized integral Radon transform. The main question is to determine information about the integrand functions if the values of some integrals are known. A feature of the work of the authors of this message is an analysis of the case when the function is integrated according to hyperplanes in finite-dimensional Euclidean space, and the integrands depend not only on the variables of integration, but also on some of the variables characterizing the hyperplanes. At the same time, the number of independent variables describing known integrals are smaller than those of the unknown integrand. We consider discontinuous integrands defined specifically introduced pseudo-convex sets. A Stefan-type problem is posed about finding surfaces discontinuities of the integrand function. The work provides formulas based on the application special integro-differential operators to known data and allowing you to solve the assigned tasks.

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

Cited by 3 publications

References 29 publications

Well-defined forward operators in dynamic diffractive tensor tomography using viscosity solutions of transport equations

Well-defined forward operators in dynamic diffractive tensor tomography using viscosity solutions of transport equations

Well-defined forward operators in dynamic diffractive tensor tomography using viscosity solutions of transport equations

Inversion problem for Radon transforms defined on pseudoconvex sets

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