On the Reconstruction of Dipole Directions from Spherical Magnetic Field Measurements

TL;DR

This paper investigates the uniqueness of reconstructing dipole directions and susceptibilities from spherical magnetic field measurements, revealing conditions under which solutions are unique or non-unique, and providing methods to identify potential dipole directions.

Contribution

It introduces new criteria for uniqueness in dipole reconstruction under spatial localization and band-limited constraints, and offers polynomial-based methods to identify candidate dipole directions.

Findings

Susceptibility is uniquely determined with known dipole direction under spatial localization.

Dipole direction cannot be uniquely determined if it is unknown, but candidates can be found via polynomial zeros.

Examples demonstrate non-uniqueness and candidate identification methods.

Abstract

Reconstructing magnetizations from measurements of the generated magnetic potential is generally non-unique. The non-uniqueness still remains if one restricts the magnetization to those induced by an ambient magnetic dipole field (i.e., the magnetization is described by a scalar susceptibility and the dipole direction). Here, we investigate the situation under the additional constraint that the susceptibility is either spatially localized in a subregion of the sphere or that it is band-limited. If the dipole direction is known, then the susceptibility is uniquely determined under the spatial localization constraint while it is only determined up to a constant under the the assumption of band-limitedness. If the dipole direction is not known, uniqueness is lost again. However, we show that all dipole directions that could possibly generate the measured magnetic potential need to be zeros of a certain polynomial which can be computed from the given potential. We provide examples of non-uniqueness of the dipole direction and examples on how to find admissible candidates for the dipole direction under the spatial localization constraint.