A unified approach to inversion formulae for vector and tensor ray and radon transforms and the Natterer inequality

IF 2 2区数学 Q1 MATHEMATICS, APPLIED

Inverse Problems Pub Date : 2024-07-09 DOI:10.1088/1361-6420/ad5d0e

Alfred K Louis

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引用次数: 0

Abstract

Most derivations of inversion formulae for x-ray or Radon transform are based on the projection theorem, where for fixed direction the Fourier transform of x-ray or Radon transform is calculated and compared with the Fourier transform of the searched-for function. In contrast to this we start here off from the searched-for field, calculate its Fourier transform for fixed direction, which is now a vector or tensor field, that we then expand in a suitable direction dependent basis. The expansion coefficients are recognized as the Fourier transform of longitudinal, transversal or mixed ray transforms or vectorial Radon transform respectively. The inverse Fourier transform of the searched-for field then directly leads to inversion formulae for those transforms applying problem adapted backprojections. When considering the Helmholtz decomposition of the field we immediately find inversion formulae for those transversal or longitudinal transforms. First inversion formulae for the longitudinal ray transform, similar to those given by Natterer (1986 The Mathematics of Computerized Tomography (Teubner and Wiley)) for x-ray tomography, were given by Natterer-Wübbeling in 2001, Natterer and Wübbeling (2001 Mathematical Methods in Image Reconstruction (SIAM)), but then not pursued by other authors. In this paper, we present the above described method and derive in a unified way inversion formulae for the ray transforms treated in Louis (2022 Inverse Problems 38 065008) containing the results from Louis (2022 Inverse Problems 38 065008) as special cases. Additionally we present new inversion formulae for the vectorial Radon transform. As a consequence the inversion formulae directly give Plancherel’s formulae for the vectorial or tensorial transforms. Together with the Natterer inequality, which is independent of the ray or Radon transforms, we present the Natterer stability of those vectorial and tensorial transforms.

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矢量和张量射线与氡变换及纳特勒不等式反演公式的统一方法

大多数 X 射线或拉顿变换反演公式的推导都是基于投影定理，即在固定方向上计算 X 射线或拉顿变换的傅里叶变换，并将其与搜索函数的傅里叶变换进行比较。与此相反，我们在这里从搜索到的场开始，计算其固定方向的傅里叶变换，现在它是一个矢量或张量场，然后我们在一个合适的与方向相关的基础上对其进行扩展。展开系数分别被视为纵向、横向或混合射线变换的傅里叶变换或矢量拉顿变换。搜索场的反傅里叶变换可直接得出这些变换的反演公式，并应用与问题相适应的反推。在考虑场的亥姆霍兹分解时，我们可以立即找到这些横向或纵向变换的反演公式。Natterer-Wübbeling 于 2001 年给出了纵向射线变换的反演公式，Natterer 和 Wübbeling (2001 Mathematical Methods in Image Reconstruction (SIAM))也给出了与 Natterer（1986 The Mathematics of Computerized Tomography (Teubner and Wiley)）类似的 X 射线断层摄影反演公式，但其他作者并未继续研究。在本文中，我们介绍了上述方法，并以统一的方式推导出路易斯（2022 逆问题 38 065008）中处理的射线变换的反演公式，其中包含路易斯（2022 逆问题 38 065008）中作为特例的结果。此外，我们还提出了矢量拉顿变换的新反演公式。因此，反演公式直接给出了矢量或张量变换的 Plancherel 公式。通过与射线或拉顿变换无关的纳特勒不等式，我们提出了这些矢量和张量变换的纳特勒稳定性。

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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.