二维矢量层析成像中的广义V线变换
Generalized V-line transforms in 2D vector tomography
摘要：
我们研究了从一组新的广义$V$ -线变换中恢复$\mathbb{R}^2$中向量场的反问题。首先介绍了$\mathbb{R}^2$中向量场的纵向和横向$V$ -线变换.然后，我们给出了一个明确的表征，他们各自的内核，并表明，他们是互补的。我们证明可逆的每个变换模他们的内核，并结合它们来重建明确的完整的向量场。在第二种方法中，我们将纵向和横向V线变换与它们相应的一阶矩变换相结合，并从任一对中恢复完整的向量场。我们表明，在这些设置中的每一个可用的数据可以用来推导出的向量场的标量分量的有符号的V线变换，并使用后者的已知的反转。本文的最后一个主要结果是推导出了一个精确的封闭形式公式，用于从$\mathbb{R}^2$的带权星变换重构其全向量场。我们通过将向量场的星形变换与场的标量分量的普通Radon变换相关联来解决这个问题。
英文摘要：
We study the inverse problem of recovering a vector field in $\mathbb{R}^2$ from a set of new generalized $V$-line transforms in three different ways. First, we introduce the longitudinal and transverse $V$-line transforms for vector fields in $\mathbb{R}^2$. We then give an explicit characterization of their respective kernels and show that they are complements of each other. We prove invertibility of each transform modulo their kernels and combine them to reconstruct explicitly the full vector field. In the second method, we combine the longitudinal and transverse V-line transforms with their corresponding first moment transforms and recover the full vector field from either pair. We show that the available data in each of these setups can be used to derive the signed V-line transform of both scalar component of the vector field, and use the known inversion of the latter. The final major result of this paper is the derivation of an exact closed form formula for reconstruction of the full vector field in $\mathbb{R}^2$ from its star transform with weights. We solve this problem by relating the star transform of the vector field to the ordinary Radon transform of the scalar components of the field.
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