X-ray tomography of one-forms with partial data

TL;DR

This paper proves that under certain conditions, a closed one-form with vanishing line integrals over lines meeting a small set must be exact everywhere, using unique continuation and partial data analysis.

Contribution

It introduces new unique continuation results for the X-ray transform of one-forms with partial data, applicable to compactly supported covector distributions.

Findings

Vanishing line integrals imply exactness of the one-form under specified conditions

Established unique continuation for the normal operator of the X-ray transform

Provided two proofs for the partial data result

Abstract

If the integrals of a one-form over all lines meeting a small open set vanish and the form is closed in this set, then the one-form is exact in the whole Euclidean space. We obtain a unique continuation result for the normal operator of the X-ray transform of one-forms, and this leads to one of our two proofs of the partial data result. Our proofs apply to compactly supported covector-valued distributions.