Inversion of generalized Radon transforms acting on 3D vector and symmetric tensor fields

IF 2 2区数学 Q1 MATHEMATICS, APPLIED

Inverse Problems Pub Date : 2023-12-05 DOI:10.1088/1361-6420/ad0fac

Ivan E Svetov, Anna P Polyakova

{"title":"Inversion of generalized Radon transforms acting on 3D vector and symmetric tensor fields","authors":"Ivan E Svetov, Anna P Polyakova","doi":"10.1088/1361-6420/ad0fac","DOIUrl":null,"url":null,"abstract":"Currently, theory of the ray transforms of vector and tensor fields is well developed, but the generalized Radon transforms of such fields have not been fully studied. We consider the normal, longitudinal and mixed Radon transforms (with integration over planes) acting on three-dimensional vector and symmetric tensor fields. We prove that these operators are continuous. In case when values of all generalized Radon transforms are known, inversion formulas are derived for componentwise reconstruction of vector and symmetric <italic toggle=\"yes\">m</italic>-tensor fields, <inline-formula>\n<tex-math><?CDATA $m\\unicode{x2A7E}2$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mi>m</mml:mi><mml:mtext>⩾</mml:mtext><mml:mn>2</mml:mn></mml:math>\n<inline-graphic xlink:href=\"ipad0facieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>. Novel detailed decompositions of 3D vector and symmetric 2-tensor fields as a sum of pairwise orthogonal terms are obtained. For construction of each term in the sum only one function is required. With usage of these decompositions we have described the kernels and images of the generalized Radon transforms. In addition, we have obtained inversion formulas for each of the generalized Radon transforms acting on 3D vector and symmetric 2-tensor fields and have formulated theorems similar to the projection theorem for the Radon transform. For the cases <inline-formula>\n<tex-math><?CDATA $m\\unicode{x2A7E}3$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mi>m</mml:mi><mml:mtext>⩾</mml:mtext><mml:mn>3</mml:mn></mml:math>\n<inline-graphic xlink:href=\"ipad0facieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> similar statements are formulated as hypotheses. In addition, we consider the weighted longitudinal Radon transforms of 3D vector fields. Formulas are obtained for reconstructing the potential part of a 3D vector field from the known values of the longitudinal Radon transforms and one weighted Radon transform. Finally, we discuss the problem of vector fields reconstruction in <inline-formula>\n<tex-math><?CDATA $\\mathbb{R}^n$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math>\n<inline-graphic xlink:href=\"ipad0facieqn3.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>, <inline-formula>\n<tex-math><?CDATA $n\\unicode{x2A7E}4$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mi>n</mml:mi><mml:mtext>⩾</mml:mtext><mml:mn>4</mml:mn></mml:math>\n<inline-graphic xlink:href=\"ipad0facieqn4.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"21 1","pages":""},"PeriodicalIF":2.0000,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Inverse Problems","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1088/1361-6420/ad0fac","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

Currently, theory of the ray transforms of vector and tensor fields is well developed, but the generalized Radon transforms of such fields have not been fully studied. We consider the normal, longitudinal and mixed Radon transforms (with integration over planes) acting on three-dimensional vector and symmetric tensor fields. We prove that these operators are continuous. In case when values of all generalized Radon transforms are known, inversion formulas are derived for componentwise reconstruction of vector and symmetric m-tensor fields,

m⩾2

. Novel detailed decompositions of 3D vector and symmetric 2-tensor fields as a sum of pairwise orthogonal terms are obtained. For construction of each term in the sum only one function is required. With usage of these decompositions we have described the kernels and images of the generalized Radon transforms. In addition, we have obtained inversion formulas for each of the generalized Radon transforms acting on 3D vector and symmetric 2-tensor fields and have formulated theorems similar to the projection theorem for the Radon transform. For the cases

m⩾3

similar statements are formulated as hypotheses. In addition, we consider the weighted longitudinal Radon transforms of 3D vector fields. Formulas are obtained for reconstructing the potential part of a 3D vector field from the known values of the longitudinal Radon transforms and one weighted Radon transform. Finally, we discuss the problem of vector fields reconstruction in

n⩾4

查看原文本刊更多论文

作用于三维向量场和对称张量场的广义拉顿变换的反演

目前，矢量场和张量场的射线变换理论已经发展成熟，但此类场的广义拉顿变换尚未得到充分研究。我们考虑了作用于三维向量场和对称张量场的法向、纵向和混合拉顿变换（在平面上积分）。我们证明这些算子是连续的。在已知所有广义拉顿变换值的情况下，推导出了反演公式，用于分量重构矢量和对称 m 张量场 m⩾2。将三维矢量场和对称 2 张量场分解为成对正交项的总和，获得了新的详细分解。和中每个项的构造只需要一个函数。利用这些分解，我们描述了广义拉顿变换的核和图像。此外，我们还获得了作用于三维向量场和对称 2 张量场的广义拉顿变换的反演公式，并提出了与拉顿变换投影定理类似的定理。对于 m⩾3 的情况，也提出了类似的假设。此外，我们还考虑了三维向量场的加权纵向拉顿变换。根据已知的纵向 Radon 变换值和一个加权 Radon 变换，我们可以得到重建三维矢量场势能部分的公式。最后，我们讨论了 Rn, n⩾4 中的向量场重建问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Inverse Problems 数学-物理：数学物理

CiteScore

4.40

自引率

14.30%

发文量

115

审稿时长

2.3 months

期刊介绍： An interdisciplinary journal combining mathematical and experimental papers on inverse problems with theoretical, numerical and practical approaches to their solution. As well as applied mathematicians, physical scientists and engineers, the readership includes those working in geophysics, radar, optics, biology, acoustics, communication theory, signal processing and imaging, among others. The emphasis is on publishing original contributions to methods of solving mathematical, physical and applied problems. To be publishable in this journal, papers must meet the highest standards of scientific quality, contain significant and original new science and should present substantial advancement in the field. Due to the broad scope of the journal, we require that authors provide sufficient introductory material to appeal to the wide readership and that articles which are not explicitly applied include a discussion of possible applications.