A Fourier transform technique for estimating bioelectric currents from magnetic field measurements.

Abstract

This dissertation presents a new noniterative technique for estimating current densities from magnetic field measurements. This Fourier-transform technique starts by forming a set of linear equations from the Fourier-transformed Maxwell equations. The set of equations is sampled according to the Whittaker-Shannon sampling theorem and solved by matrix methods. Two variations of the technique are extensions of a Fourier-transform method developed by Dallas. The first method assumes that the x- and y-components of the magnetic field in a forbidden region are zero, and that the z-component of the current density is zero. The second method assumes that ∇²Bₓ and ∇²B(y) are zero in a forbidden region, and is not restricted to reconstructing currents with zero z-component. The assumptions about the current distribution and measurement geometry are included in the reconstruction technique by means of the sampling theorem. The effects on the reconstructions of the spatial sampling requirements and the form approximations to the differentiation operator were investigated for the two variations of the reconstruction method. We tested the Fourier-transform reconstruction methods on computer-simulated magnetic fields derived from analytic expressions. The results of the computer simulations were confirmed by reconstructions from measured magnetic fields due to known current sources. The measured magnetic fields were due to both widely distributed and highly localized current sources. If the magnetic field was undersampled, then the reconstructed current was a larger size than it should have been. The form of the differentiation operator made a dramatic difference in the reconstructed current, and in some cases resulted in a current that was not physical. Both reconstruction methods were able to distinguish between large and small current densities, had excellent lateral resolution, but were unable to provide any information about the depth of the source.