On the Degree of Ill-Posedness of Multi-Dimensional Magnetic Particle Imaging

TL;DR

This paper investigates the mathematical ill-posedness of multi-dimensional magnetic particle imaging by analyzing the singular value decay of the integral operator in the equilibrium model, considering experimental parameters.

Contribution

It provides a detailed analysis of the degree of ill-posedness in magnetic particle imaging's equilibrium model, including effects of experimental parameters and magnetic field configurations.

Findings

Singular value decay estimates reveal the ill-posedness degree.

Field free point and line configurations influence problem stability.

Numerical experiments support theoretical analysis.

Abstract

Magnetic particle imaging is an imaging modality of relatively recent origin, and it exploits the nonlinear magnetization response for reconstructing the concentration of nanoparticles. Since first invented in 2005, it has received much interest in the literature. In this work, we study one prototypical mathematical model in multi-dimension, i.e., the equilibrium model, which formulates the problem as a linear Fredholm integral equation of the first kind. We analyze the degree of ill-posedness of the associated linear integral operator by means of the singular value decay estimate for Sobolev smooth bivariate functions, and discuss the influence of various experimental parameters. In particular, applied magnetic fields with a field free point and a field free line are distinguished. The study is complemented with extensive numerical experiments.