Inversion of the two-data spherical Radon transform with the centers on a plane

IF 1.2 3区数学 Q1 MATHEMATICS

Journal of Mathematical Analysis and Applications Pub Date : 2025-03-21 DOI:10.1016/j.jmaa.2025.129512

Rafik Aramyan

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引用次数: 0

Abstract

Hyperplane is a set of non-injectivity of the spherical Radon transform (SRT) in

R^{d}

. It is possible to reconstruct a function compactly supported on one side of a hyperplane using SRT over spheres centered on the hyperplane. In this article, for the reconstruction of

f \in C (R^{3})

(the support can be non-compact) using SRT over spheres centered on a plane, an additional condition is found, which is a weighted SRT (to reconstruct an odd function with respect to the hyperplane), and the injectivity of the so-called two data spherical Radon transform is considered. The transform consists of the classical SRT and the weighted SRT. An inversion formula of the transform that uses the local data of the spherical integrals to reconstruct the unknown function is presented. The inversion formula generalizes the inversion formula of SRT for functions supported on one side of a plane, as obtained earlier by the author of this article. Such inversions have theoretical significance in many areas of mathematics and are the mathematical base of modern modalities of imaging, such as Thermo and photoacoustic tomography, radar imaging, geophysics, and a few others.

查看原文本刊更多论文

以平面为中心的双数据球面Radon变换的反演

超平面是球面Radon变换（SRT）在Rd中的非注入性集合。利用SRT在以超平面为中心的球体上重建一个紧支撑在超平面一侧的函数是可能的。本文利用以平面为中心的球面上的SRT对f∈C(R3)（支持可以是非紧致的）进行重构，发现了一个附加条件，即加权SRT（关于超平面重建奇函数），并考虑了所谓的两数据球面Radon变换的注入性。该变换由经典SRT和加权SRT组成。给出了利用球面积分的局部数据重构未知函数的反演公式。该反演公式推广了本文作者之前得到的单侧支撑函数的SRT反演公式。这种反演在许多数学领域具有理论意义，并且是现代成像方式的数学基础，例如热成像和光声层析成像、雷达成像、地球物理学和其他一些方法。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Mathematical Analysis and Applications 数学-数学

CiteScore

2.50

自引率

7.70%

发文量

790

审稿时长

6 months

期刊介绍： The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions. Papers are sought which employ one or more of the following areas of classical analysis: • Analytic number theory • Functional analysis and operator theory • Real and harmonic analysis • Complex analysis • Numerical analysis • Applied mathematics • Partial differential equations • Dynamical systems • Control and Optimization • Probability • Mathematical biology • Combinatorics • Mathematical physics.