Ellipsoidal head model and reduced-rank beamforming for EEG/MEG source estimation

We address the problems of modeling and analysis of fetal magnetoencephalographic (fMEG) data, and reduced-rank spatial filtering (beamforming) for electroencephalography (EEG) and magnetoencephalography (MEG).; The first part of this thesis is dedicated to fMEG, which is a non-invasive technique where measurements of the magnetic field outside the maternal abdomen are used to infer the source location and signals of the fetus' neural activity. We propose a solution to the forward problem of fMEG based on an ellipsoidal head geometry, which provides a more realistic approximation to the fetal head geometry compared with commonly used spherical models. We show that the ellipsoidal head model has the advantage of highlighting special characteristics of the field that are inherent to the anisotropy of the human head, such as the spread and orientation of the field as functions of the localization and position of the fetal head. Our forward solution is presented in the form of a kernel matrix that facilitates the solution of the inverse problem by decoupling the dipole localization parameters from the source signals. Furthermore, we introduce a method to obtain accurate estimates of the parameters of the ellipsoid (center location and length of semiaxes) through a least-squares fit on anatomical information obtained from 3-dimesional ultrasound images. For the inverse problem in fMEG, we use the maximum likelihood (ML) technique to estimate the location and the source signal components when measurements at multiple trials are available. We evaluate the accuracy of the ML technique by computing the bias of the source localization estimates as a function of the signal-to-noise ratio (SNR) and number of independent trials available.; In the second part of this thesis, we study the performance of various linearly constrained minimum variance (LCMV) beamformers for estimating a current dipole source at a known location using EEG/MEG data. We present our beamformers in the equivalent unconstrained form of the generalized sidelobe canceler (GSC). Under this structure, the LCMV filtering problem can be solved by finding the filter that achieves the minimum mean-squared error (MMSE) between the mainbeam response and filtered observed signal. We express the MMSE as a function of the filter's rank and use it as a criterion to evaluate the performance of the beamformers. We do not make any assumptions on the ranks of the interference-plus-noise covariance or constraint matrices. Instead, we treat them as low-rank and derive a general expression for the MMSE. We present numerical examples to compare the performances of four LCMV beamformers commonly studied in the literature: principal components, cross-spectral metrics, diagonal loading, and eigencanceler beamformers. For each of these beamformers, we show the performance in terms of their MMSE, output mean-squared error (MSE), and output SNR.