Reconstruction of the Singular Support of a Tensor Field Given in a Refracting Medium by Its Ray Transform

E. Yu. Derevtsov; S. V. Maltseva

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Reconstruction of the Singular Support of a Tensor Field Given in a Refracting Medium by Its Ray Transform

机译：通过其射线变换重建折射介质中给出的张量场的奇异支撑

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We propose some approaches for numerically solving the problem of reconstructing the singular support of a symmetric tensor field, given in a refracting medium, by its known ray transform. To solve the problem, we use the back-projection operators that act on the ray transforms and the tensor analysis methods on Riemannian manifolds. We construct the operators of the medium inhomogeneity indicator that allow us to identify the sets of points of the singular support of the scalar, vector, and tensor fields. We propose and implement algorithms for solving the problem under study.

机译：我们提出了一些方法来通过已知的射线变换来数值解决在折射介质中给定的对称张量场的奇异支撑的重构问题。为了解决该问题，我们使用了作用于射线变换的反投影算子和黎曼流形上的张量分析方法。我们构造中等不均匀性指标的算子，使我们能够识别标量，矢量和张量场的奇异支持点集。我们提出并实施用于解决所研究问题的算法。

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《Journal of applied and industrial mathematics》 |2015年第4期|共14页
作者
E. Yu. Derevtsov; S. V. Maltseva;
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原文格式 PDF
正文语种 eng
中图分类应用数学;
关键词
tomography; tensor field; function discontinuity; singular support; refraction; ray transform; back-projection operator; tensor analysis;

机译：体层摄影术;张量场;功能不连续;奇异支持;折射;射线变换;反投影算子;张量分析;

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1. Reconstruction of the Singular Support of a Tensor Field Given in a Refracting Medium by Its Ray Transform [J] . E. Yu. Derevtsov, S. V. Maltseva Journal of applied and industrial mathematics . 2015,第4期

机译：通过其射线变换重建折射介质中给出的张量场的奇异支撑
2. Reconstruction of Vector Fields and Their Singularities from Ray Transforms [J] . E. Yu. Derevtsov, V. V. Pickalov Numerical analysis and applications . 2011,第1期

机译：从射线变换重构矢量场及其奇异性
3. Support theorems for the transverse ray transform of tensor fields of rank m [J] . Journal of Mathematical Analysis and Applications . 2020,第2期

机译：支持卷曲横谱面横向射线变换的定理
4. Differential Equations and Uniqueness Theorems for the Generalized Attenuated Ray Transforms of Tensor Fields [C] . Evgeny Yu. Derevtsov, Yuriy S. Volkov, Thomas Schuster International conference on mumerical computations: theory and algorithms . 2019

机译：张量场的广义衰减射线变换的微分方程和唯一性定理
5. The X-ray transform of tensor fields. [D] . Chappa, Eduardo Enrique. 2002

机译：张量场的X射线变换。
6. Simultaneous Maximum-Likelihood Reconstruction of Absorption Coefficient Refractive Index and Dark-Field Scattering Coefficient in X-Ray Talbot-Lau Tomography [O] . André Ritter, Gisela Anton, Thomas Weber -1

机译：X射线Talbot-Lau层析成像中吸收系数折射率和暗场散射系数的同时最大似然重建
7. A support theorem for the geodesic ray transform of symmetric tensor fields [O] . Krishnan, Venky, Stefanov, Plamen 2008

机译：对称张量的测地线变换的支持定理领域
8. Refracted-Ray Scanning (Refracted Near-Field Scanning) for Measuring Index Profiles of Optical Fibers [R] . Young, M. 1981

机译：折射光扫描（折射近场扫描）用于测量光纤的折射率分布

Reconstruction of the Singular Support of a Tensor Field Given in a Refracting Medium by Its Ray Transform

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