Mathematical Analysis of the 1D Model and Reconstruction Schemes for Magnetic Particle Imaging

TL;DR

This paper provides a mathematical analysis of the MPI operator in one dimension, revealing its ill-posedness and exponential decay of singular values, supported by numerical studies.

Contribution

It introduces a Hilbert space framework for analyzing the MPI operator and characterizes its mathematical properties and ill-posedness in the univariate case.

Findings

MPI operator's singular values decay exponentially

The MPI forward operator is ill-posed

Numerical studies confirm rapid decay of singular values

Abstract

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.