Inversion of a restricted transverse ray transform with sources on a curve

Abstract

In this paper, a restricted transverse ray transform acting on vector and symmetric $m$-tensor fields is studied. We developed inversion algorithms using restricted transverse ray transform data to recover symmetric $m$-tensor fields in ${\mathbb{R}}^3$ and vector fields in ${\mathbb{R}}^n$. We restrict the transverse ray transform to all lines going through a fixed curve $\gamma$ that satisfies the Kirillov-Tuy condition. We show that the known restricted data can be used to reconstruct a specific weighted Radon transform of the unknown vector/tensor field's components, which we then use to explicitly recover the unknown field.