Partial data problems in scalar and vector field tomography

TL;DR

This paper establishes that partial line integral data uniquely determines scalar and vector fields under certain PDE conditions, using fractional Laplacian properties to prove a unique continuation principle.

Contribution

It introduces a novel unique continuation result for fractional Laplacians that ensures uniqueness in partial data tomography problems for scalar and vector fields.

Findings

Unique determination of scalar fields from partial line integrals.

Extension of results to vector field tomography.

Proof of unique continuation property for fractional Laplacians.

Abstract

We prove that if $P(D)$ is some constant coefficient partial differential operator and $f$ is a scalar field such that $P(D)f$ vanishes in a given open set, then the integrals of $f$ over all lines intersecting that open set determine the scalar field uniquely everywhere. This is done by proving a unique continuation property of fractional Laplacians which implies uniqueness for the partial data problem. We also apply our results to partial data problems of vector fields.