Efficient measurement and representation of magnetic fields in tomographic imaging using ellipsoidal harmonics

Given the pivotal role of magnetic fields in modern medicine, there is an increasing necessity for a precise characterization of their strength and orientation at high spatial and temporal resolution. As source-free magnetic fields present in tomographic imaging can be described by harmonic polynomials, they can be efficiently represented using spherical harmonic expansions, which allows for measurement at a small set of points on a sphere surrounding the field of view. However, the majority of closed-bore systems possess a cylindrical field of view, making a sphere an inadequate choice for coverage. Ellipsoids represent a superior geometrical choice, and the theory of ellipsoidal harmonic expansions can be applied to magnetic fields in an analogous manner. Despite the mathematical principles underpinning ellipsoidal harmonics being well-established, their utilization in practical applications remains relatively limited. In this study, we present an effective and flexible approach to measuring and representing magnetic fields present in tomographic imaging, which draws upon the theory of ellipsoidal harmonic expansions.