Tensor tomography using V-line transforms with vertices restricted to a circle

Abstract

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) integral moment longitudinal/transverse V-line transforms.