On the reconstruction of inducing dipole directions and susceptibilities from knowledge of the magnetic field on a sphere. Issue 1 (2nd January 2019)

• Record Type: Journal Article
• Authors: Gerhards, Christian
• Is Part Of: Inverse problems in science and engineering. Volume 27:Issue 1(2019)
• Page Start: 37
• Page End: 60
• Publication Date: 2019-01-02
• Subjects: Inverse magnetization problem -- decomposition of spherical vector fields -- uniqueness -- magnetic dipoles -- susceptibility; 45Q05 -- 33C55 -- 86A22 -- 86A25; Engineering mathematics -- Periodicals; Inverse problems (Differential equations) -- Periodicals; 620.001515357
• DOI: 10.1080/17415977.2018.1438426
• Languages: English
• ISSNs: 1741-5977

Abstract

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 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.