Model-Based Reconstruction for Magnetic Particle Imaging in 2D and 3D

TL;DR

This paper develops mathematical models and explicit reconstruction formulae for magnetic particle imaging in 2D and 3D, addressing the lack of such formulae beyond 1D and analyzing the problem's ill-posedness.

Contribution

It introduces novel 2D and 3D reconstruction formulae for MPI, extending beyond previous measurement-based methods and analyzing the problem's ill-posed nature.

Findings

Derived explicit 2D and 3D reconstruction formulae

Identified the severe ill-posedness of MPI reconstruction

Developed a model-based reconstruction algorithm

Abstract

We contribute to the mathematical modeling and analysis of magnetic particle imaging which is a promising new in-vivo imaging modality. Concerning modeling, we develop a structured decomposition of the imaging process and extract its core part which we reveal to be common to all previous contributions in this context. The central contribution of this paper is the development of reconstruction formulae for MPI in 2D and 3D. Until now, in the multivariate setup, only time consuming measurement approaches are available, whereas reconstruction formulae are only available in 1D. The 2D and the 3D (describing the real world) reconstruction formulae which we derive here are significantly different from the 1D situation -- in particular there is no Dirac property in dimensions greater than one when the particle sizes approach zero. As a further result of our analysis, we conclude that the reconstruction problem in MPI is severely ill-posed. Finally, we obtain a model-based reconstruction algorithm.