International Journal on Magnetic Particle Imaging IJMPI
Vol. 12 No. 1 Suppl 1 (2026): Int J Mag Part Imag
https://doi.org/10.18416/IJMPI.2026.2603032

Proceedings Articles, ID 927

Coordinate transformations for magnetic fields’ solid spherical harmonic coefficients

Main Article Content

Marija Boberg (1) Section for Biomedical Imaging, University Medical Center Hamburg-Eppendorf, Hamburg, Germany; 2) Institute for Biomedical Imaging, Hamburg University of Technology, Hamburg, Germany), Fynn Foerger (1) Section for Biomedical Imaging, University Medical Center Hamburg-Eppendorf, Hamburg, Germany; 2) Institute for Biomedical Imaging, Hamburg University of Technology, Hamburg, Germany), Tobias Knopp (1) Section for Biomedical Imaging, University Medical Center Hamburg-Eppendorf, Hamburg, Germany; 2) Institute for Biomedical Imaging, Hamburg University of Technology, Hamburg, Germany; 3) Fraunhofer Research Institute for Individualized and Cell-based Medicine, IMTE, Lübeck, Germany), Martin Möddel (1) Section for Biomedical Imaging, University Medical Center Hamburg-Eppendorf, Hamburg, Germany; 2) Institute for Biomedical Imaging, Hamburg University of Technology, Hamburg, Germany)

Copyright (c) 2026 Marija Boberg, Fynn Foerger, Tobias Knopp, Martin Möddel

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

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.

Article Details

References

[1] M. Boberg, T. Knopp, and M. Möddel. Unique compact representation of magnetic fields using truncated solid harmonic expansions. European Journal of Applied Mathematics, pp. 1–28, 2025, doi:10.1017/S0956792524000883.
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[3] E. O. Steinborn and K. Ruedenberg, Rotation and Translation of Regular and Irregular Solid Spherical Harmonics, in Advances in Quantum Chemistry, P.- O. Löwdin, Ed., 7, Academic Press, 1973, 1–81. doi:10.1016/S0065-3276(08)60558-4.
[4] F. Thieben, F. Foerger, F. Mohn, N. Hackelberg, M. Boberg, J.- P. Scheel, M. Möddel, M. Graeser, and T. Knopp. System characterization of a human-sized 3D real-time magnetic particle imaging scanner for cerebral applications. Communications Engineering, 3(1):1–17, 2024, doi:10.1038/s44172-024-00192-6.

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