On the degree of ill-posedness of multi-dimensional magnetic particle imaging

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 being 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 on the decay rate. 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.