In this article, we study the problem of recovering symmetric $m$-tensor fields (including vector fields) supported in a unit disk $\mathbb{D}$ from a set of generalized V-line transforms, namely longitudinal, transverse, and mixed V-line transforms, and their integral moments. We work in a circular geometric setup, where the V-lines have vertices on a circle, and the axis of symmetry is orthogonal to the circle. We present two approaches to recover a symmetric $m$-tensor field from the combination of longitudinal, transverse, and mixed V-line transforms. With the help of these inversion results, we are able to give an explicit kernel description for these transforms. We also derive inversion algorithms to reconstruct a symmetric $m$-tensor field from its first $(m+1)$ moment longitudinal/transverse V-line transforms.

翻译：本文研究从一组广义V线变换（即纵向、横向及混合V线变换）及其积分矩中恢复单位圆盘$\mathbb{D}$内对称$m$阶张量场（包括矢量场）的问题。我们采用圆形几何构型，其中V线顶点位于圆周上，且对称轴与圆周正交。我们提出两种方法，通过结合纵向、横向及混合V线变换来恢复对称$m$阶张量场。借助这些反演结果，我们能够明确给出这些变换的核描述。此外，我们还推导了从对称$m$阶张量场的前$(m+1)$阶矩纵向/横向V线变换中重建该张量场的反演算法。