论文标题
2D矢量断层扫描中的广义V线变换
Generalized V-line transforms in 2D vector tomography
论文作者
论文摘要
我们研究了从一组新的通用$ V $ -line以三种不同方式转换的$ \ mathbb {r}^2 $中恢复向量字段的反问题。首先，我们介绍了$ \ mathbb {r}^2 $中的向量字段的纵向和横向$ V $ - 线变换。然后，我们给出了它们各自内核的明确表征，并表明它们是彼此的补充。我们证明了每个变换模量的可逆性，它们的内核并将它们组合起来以明确重建完整的向量字段。在第二种方法中，我们将纵向和横向V线变换及其相应的第一刻变化结合在一起，并从任一对中恢复完整的矢量场。我们表明，这些设置中的每个设置中的可用数据可用于得出矢量字段两个标量组件的符号V-Line变换，并使用后者的已知反转。本文的最终主要结果是从其恒星转换中的$ \ Mathbb {r}^2 $重建的精确封闭形式公式，以重建整个矢量字段。我们通过将矢量场的恒星变换与该场标量分量的普通rad变换联系起来来解决此问题。
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.
