Full reconstruction of a vector field from restricted Doppler and first integral moment transforms in ${\mathbb{R}}^n$

Abstract

We show that a vector field in ${\mathbb{R}}^n$ can be reconstructed uniquely from the knowledge of restricted Doppler and first integral moment transforms. The line complex we consider consists of all lines passing through a fixed curve $\gamma \subset {\mathbb{R}}^n$. The question of reconstruction of a symmetric $m$-tensor field from the knowledge of the first $m+1$ integral moments was posed by Sharafutdinov in his book (see pp. 78), "Integral geometry of tensor fields," Inverse and Ill-posed problems series, De Grutyer. In this work, we provide an answer to Sharafutdinov's question for the case of vector fields from restricted data comprising of the first $2$ integral moment transforms.