A detailed study on the use of polynomial functions for modeling geometric distortion in magnetic resonance imaging

The use of polynomial functions for modeling geometric distortion in magnetic resonance imaging (MRI) that arises from scanner's hardware imperfection is studied in detail. In this work, the geometric distortion data from four representative MRI systems were used. Modeling of these data using polynomial functions of the fourth, fifth, sixth, and seventh orders was carried out. In order to investigate how this modeling performed for different size and shape of the volume of interest, the modeling was carried out for three different volumes of interest (VOI): a cube, a cylinder, and a sphere. The modeling's goodness was assessed using both the maximum and mean absolute errors. The modeling results showed that (i) for the cube VOI there appears to be an optimal polynomial function that gives the least modeling errors and the sixth order polynomial was found to be the optimal polynomial function for the size of the cubic VOI considered in the present work; (ii) for the cylinder VOI, all four polynomials performed approximately equally well but a trend of a slight decrease in the mean absolute error with the increasing order of the polynomial was noted; and (iii) for the sphere VOI, the maximum absolute error showed some variations with the order of the polynomial, with the fourth order polynomial producing the smallest maximum absolute errors. It is further noted that extrapolation could lead to very large errors so any extrapolation needs to be avoided. A detailed analysis on the modeling errors is presented.