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

TL;DR

This paper demonstrates that a vector field in b^n can be uniquely reconstructed from restricted Doppler and first integral moment transforms along lines passing through a fixed curve, addressing a question posed by Sharafutdinov.

Contribution

It provides a novel solution for reconstructing vector fields from limited integral data, specifically the first two integral moment transforms, in the context of restricted line complexes.

Findings

Unique reconstruction of vector fields from restricted transforms.

Addresses Sharafutdinov's question for vector fields.

Provides explicit reconstruction methods from limited data.

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.