Numerical solution of an inverse problem in magnetic resonance imaging using a regularized higher-order boundary element method

Abstract

We investigate the reconstruction of a divergence-free surface current distribution from knowledge of the magnetic flux density in a prescribed region of interest in the framework of static electromagnetism. This inverse problem is motivated by the design of gradient coils used in magnetic resonance imaging (MRI) and is formulated using its corresponding integral representation according to potential theory. A novel higher-order boundary element method (BEM) which satisfies the continuity equation for the current density, i.e. divergence-free BEM, is also presented. Since the discretised BEM system is ill-posed and hence the associated least-squares solution may be inaccurate and/or physically meaningless, the Tikhonov regularization method is employed in order to retrieve accurate and physically correct solutions.