International Journal on Magnetic Particle Imaging IJMPI
Vol. 6 No. 1 (2020): Int J Mag Part Imag
https://doi.org/10.18416/IJMPI.2019.1912001

Research Articles

On the Representation of Magnetic Particle Imaging in Fourier Space

Main Article Content

Marco Maass (Institute for Signal Processing, University of Lübeck), Alfred Mertins (Institute for Signal Processing, University of Lübeck)

Abstract

Magnetic particle imaging is a tracer-based medical imaging modality. Although various reconstruction methods are known, such as the ones based on a measured system matrix, the mathematical formulation of physical models of magnetic particle imaging is still lacking in several ways. Even for fairly simplified models, such as the Langevin model of paramagnetism, many properties are unproven. Only when one-dimensional excitation is used, the existing models are sufficient to derive simple and fast reconstruction techniques, like the so-called x-space and Chebyshev reconstruction approaches. Recently, an accurate formulation of the one-dimensional Fourier transformof the Langevin function and related functions has been provided. The present article extends the theory to multidimensional magnetic particle imaging. The derived formulations help us to calculate the exact relationship between the system function of Lissajous field-free-point trajectory based magnetic particle imaging and tensor products of Chebyshev polynomials and also uncover a direct relationship to tensor products of Bessel functions of first kind in the spatio-temporal Fourier domain. Moreover, the developed formulation consolidates the mathematical description of magnetic particle imaging and lays the basis for the investigation of different trajectories.


 


Int. J. Mag. Part. Imag. 6(1), 2019, Article ID: 1912001, DOI: 10.18416/IJMPI.2019.1912001

Article Details

References

[1] T. Kluth. Mathematical models for magnetic particle imaging. Inverse Problems, 34(8):083001, 2018, doi:10.1088/1361-6420/aac535.

[2] J. Rahmer, J.Weizenecker, B. Gleich, and J. Borgert. Signal encoding in magnetic particle imaging: properties of the system function. BMC Medical Imaging, 9:4, 2009, doi:10.1186/1471-2342-9-4.

[3] P.W. Goodwill and S. M. Conolly. Multidimensional X-Space Magnetic Particle Imaging. IEEE Transactions on Medical Imaging, 30(9):1581–1590, 2011, doi:10.1109/TMI.2011.2125982.

[4] J. J. Konkle, P. W. Goodwill, O. M. Carrasco-Zevallos, and S. M. Conolly. Projection Reconstruction Magnetic Particle Imaging. IEEE Transactions on Medical Imaging, 32(2):338–347, 2013, doi:10.1109/TMI.2012.2227121.

[5] T. Knopp, S. Biederer, T. F. Sattel, M. Erbe, and T. M. Buzug. Prediction of the spatial resolution of magnetic particle imaging using the modulation transfer function of the imaging process. IEEE Transactions on Medical Imaging, 30(6):1284–1292, 2011, doi:10.1109/TMI.2011.2113188.

[6] M. Maass and A. Mertins, On the formulation of the magnetic particle imaging system function in Fourier space, in International Workshop on Magnetic Particle Imaging, 39–40, 2018.

[7] W. Erb, A. Weinmann, M. Ahlborg, C. Brandt, G. Bringout, T. M. Buzug, J. Frikel, C. Kaethner, T. Knopp, T. März, M. Möddel, M. Storath, and A.Weber. Mathematical analysis of the 1D model and reconstruction schemes for magnetic particle imaging. Inverse Problems, 34(5):055012, 2018, doi:10.1088/1361-6420/aab8d1.

[8] A. Cordes and T. M. Buzug, Deconvolution kernel for 1D x-space MPI, in International Workshop on Magnetic Particle Imaging, 49–50, 2018.

[9] T. Knopp and T. M. Buzug,Magnetic Particle Imaging: An Introduction to Imaging Principles and Scanner Instrumentation. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012, doi:10.1007/978-3-642-04199-0.

[10] P. Goodwill, G. Scott, P. Stang, and S. Conolly. Narrowband Magnetic Particle Imaging. IEEE Transactions on Medical Imaging, 28(8):1231–1237, 2009, doi:10.1109/TMI.2009.2013849.

[11] H. Schomberg, Magnetic particle imaging: Model and reconstruction, in IEEE International Symposium on Biomedical Imaging: From Nano to Macro, 992–995, IEEE, 2010. doi:10.1109/ISBI.2010.5490155.

[12] S. Bergner, T. Moller, D. Weiskopf, and D. J. Muraki. A Spectral Analysis of Function Composition and its Implications for Sampling in Direct Volume Visualization. IEEE Transactions on Visualization and Computer Graphics, 12(5):1353–1360, 2006, doi:10.1109/TVCG.2006.113.

[13] M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, AppliedMa. Washington, D.C.: National Bureau of Standards, 1964,

[14] K. A. Stromberg, An Introduction to Classical Real Analysis. American Mathematical Society Chelsea Publishing, 2015, ISBN: 1470425440.

[15] L. Grafakos and G. Teschl. On Fourier Transforms of Radial Functions and Distributions. Journal of Fourier Analysis and Applications, 19(1):167–179, 2013, doi:10.1007/s00041-012-9242-5.

[16] R. Estrada. On Radial Functions and Distributions and Their Fourier Transforms. Journal of Fourier Analysis and Applications, 20(2):301–320, 2014, doi:10.1007/s00041-013-9313-2.

[17] H.O. Be´ca. An orthogonal set based on Bessel functions of the first kind. Publikacije Elektrotehni?ckog fakulteta. SerijaMatematika i fizika, 678(715):85–90, 1980.

[18] K. Stempak. A weighted uniform Lp –estimate of Bessel functions: A note on a paper of Guo. Proceedings of the AmericanMathematical Society, 128(10):2943–2946, 2000, doi:10.1090/S0002-9939-00-05365-X.

[19] L. J. Landau. Bessel Functions: Monotonicity and Bounds. Journal of the London Mathematical Society, 61(1):197–215, 2000, doi:10.1112/S0024610799008352.

[20] T. März and A. Weinmann. Model-based reconstruction for magnetic particle imaging in 2D and 3D. Inverse Problems and Imaging, 10(4):1087–1110, 2016, doi:10.3934/ipi.2016033.

[21] H. Bateman, Higher Transcendental Functions, volume 2, A. Erdélyi, Ed.McGraw-Hill Book Company, Inc., 1953,

[22] J. Lampe, C. Bassoy, J. Rahmer, J. Weizenecker, H. Voss, B. Gleich, and J. Borgert. Fast reconstruction in magnetic particle imaging. Physics in Medicine and Biology, 57(4):1113–1134, 2012, doi:10.1088/0031-9155/57/4/1113.

[23] T. Knopp and A. Weber. Local System Matrix Compression for Efficient Reconstruction in Magnetic Particle Imaging. Advances in Mathematical Physics, 2015:1–7, 2015, doi:10.1155/2015/472818.

[24] M. Maass, K. Bente, M. Ahlborg, H. Medimagh, H. Phan, T. M. Buzug, and A. Mertins. Optimized Compression of MPI System Matrices Using a Symmetry-Preserving Secondary Orthogonal Transform. International Journal onMagnetic Particle Imaging, 2(1), 2016, doi:10.18416/IJMPI.2016.1607002.

[25] L. Schmiester, M. Möddel, W. Erb, and T. Knopp. Direct Image Reconstruction of Lissajous-Type Magnetic Particle Imaging Data Using Chebyshev-Based Matrix Compression. IEEE Transactions on Computational Imaging, 3(4):671–681, 2017, doi:10.1109/TCI.2017.2706058.

[26] R. Piessens, Hankel Transform, in Transforms and Applications Handbook, Third Edition, A. D. Poularikas, Ed., CRC Press, 2010. doi:10.1201/9781420066531-c9.