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.

中文翻译：

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二维矢量断层扫描中的广义 V 线变换

我们研究了以三种不同方式从一组新的广义 $V$-line 变换中恢复 $\mathbb{R}^2$ 中的向量场的逆问题。首先，我们介绍了 $\mathbb{R}^2$ 中向量场的纵向和横向 $V$-line 变换。然后，我们对它们各自的内核进行了明确的表征，并表明它们是相互补充的。我们证明了每个变换的可逆性，以它们的内核为模，并将它们组合起来以显式地重建完整的向量场。在第二种方法中，我们将纵向和横向 V 线变换与其对应的一阶矩变换相结合，并从任一对中恢复完整的矢量场。我们表明，这些设置中的每一个中的可用数据都可用于导出向量场的两个标量分量的有符号 V 线变换，并使用后者的已知反演。本文的最终主要结果是推导出了一个精确的闭式公式，用于重建 $\mathbb{R}^2$ 中的全向量场，从其带权重的星形变换。我们通过将矢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。