Mathematical Analysis of the 1D Model and Reconstruction Schemes for Magnetic Particle Imaging
For researchers in MPI, this analysis characterizes the ill-posedness of the forward problem, which is foundational for developing reconstruction algorithms.
The paper provides a mathematical analysis of the 1D magnetic particle imaging (MPI) operator, showing that its singular values decay exponentially, indicating severe ill-posedness. Numerical studies confirm rapid singular value decay.
Magnetic particle imaging (MPI) is a promising new in-vivo medical imaging modality in which distributions of super-paramagnetic nanoparticles are tracked based on their response in an applied magnetic field. In this paper we provide a mathematical analysis of the modeled MPI operator in the univariate situation. We provide a Hilbert space setup, in which the MPI operator is decomposed into simple building blocks and in which these building blocks are analyzed with respect to their mathematical properties. In turn, we obtain an analysis of the MPI forward operator and, in particular, of its ill-posedness properties. We further get that the singular values of the MPI core operator decrease exponentially. We complement our analytic results by some numerical studies which, in particular, suggest a rapid decay of the singular values of the MPI operator.