Tomographic Reconstruction of the Curl and Divergence of 2D Vector Fields Taking Refractions into Account
Tomographic Reconstruction of the Curl and Divergence of 2D Vector Fields Taking Refractions into Account
复制标题
考虑折射的二维矢量场旋度和发散的层析重建
DOI：
10.1137/100786770
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
T. Schuster
中科院分区：
文献类型：
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作者：
Tim Pfitzenreiter;T. Schuster
本文关注的是根据超声飞行时间测量重建二维 (2D) 矢量场的旋度和发散度的问题，同时考虑发射超声波的折射。通常，在二维矢量场断层扫描中，超声信号的折射被忽略；即，假设信号沿直线传播。我们讨论折射效应，假设超声信号沿着黎曼度量的测地线传播，该测地线与费马原理产生的折射率相关。然而，所研究的向量场仍然定义在配备欧几里德度量的 $\mathbb{R}^2$ 中的域上。我们展示了沿着测地线的射线变换和 Radon 变换之间的联系，这概括了欧几里德情况的众所周知的结果。依靠这种关系，我们开发了一个渐近重构公式，用于使用傅里叶积分算子计算矢量场的旋度和散度。本文最后进行了详细的数值测试，一方面证明了我们的方法的良好性能，另一方面显示了与假设沿直线传播的计算相比的改进。
This article is concerned with the problem of reconstructing the curl and divergence of two-dimensional (2D) vector fields from ultrasound time-of-flight measurements taking the refraction of the emitted ultrasound rays into account. Usually, in 2D vector field tomography the refraction of the ultrasound signal is neglected; i.e., one assumes that the signal propagates along straight lines. We address refractive effects, assuming that the ultrasound signal propagates along the geodesics of a Riemannian metric that is associated with the refractive index which is due to Fermat's principle. The investigated vector field, however, is still defined on a domain in $\mathbb{R}^2$ that is equipped with the Euclidean metric. We show a connection between the ray transforms and the Radon transform along geodesic curves which generalizes a well-known result from the Euclidean case. Relying on this relation, we develop an asymptotic reconstruction formula for computing the curl and divergence of the vector field using Fourier integral operators. This article concludes with detailed numerical tests proving on the one hand a good performance of our method and showing on the other hand an improvement compared with computations that assume a propagation along straight lines.