Coordinate transformations for magnetic fields’ solid spherical harmonic coefficients

Magnetic particle imaging fundamentally relies on magnetic fields, so accurate characterization of the fields is crucial for various applications, such as sequence planning or model-based reconstructions. Magnetic fields can be compactly represented by solid harmonic expansions, whose coefficients efficiently describe the field distribution. However, in practice, the coordinate systems of measurement devices are not always perfectly aligned with the scanner's coordinate system, complicating direct comparison and integration of measured data from different setups. In this work, we present methods for transforming solid harmonic coefficients under translations, rotations, and point reflections. These transformations enable consistent mapping of field representations into a common coordinate system, allowing direct comparison of magnetic fields obtained in different coordinate frames.