In this thesis, three different approaches are developed for the estimation of focal brain activity using EEG measurements. The proposed approaches have been tested and found feasible using simulated data.
First, we develop a robust solver for the recovery of focal dipole sources. The solver uses a weighted dipole strength penalty term (also called weighted L1,2 norm) as prior information in order to ensure that the sources are sparse and focal, and that both the source orientation and depth bias are reduced. The solver is based on the truncated Newton interior point method combined with a logarithmic barrier method for the approximation of the penalty term. In addition, we use a Bayesian framework to derive the depth weights in the prior that are used to reduce the tendency of the solver to favor superficial sources.
In the second approach, vector field tomography (VFT) is used for the estimation of underlying electric fields inside the brain from external EEG measurements. The electric field is
reconstructed using a set of line integrals. This is the first time that VFT has been used for the
recovery of fields when the dipole source lies inside the domain of reconstruction. The benefit
of this approach is that we do not need a mathematical model for the sources. The test cases indicated that the approach can accurately localize the source activity.
In the last part of the thesis, we show that, by using the Bayesian approximation error approach (AEA), precise knowledge of the tissue conductivities and head geometry are not
always needed. We deliberately use a coarse head model and we take the typical variations
in the head geometry and tissue conductivities into account statistically in the inverse model.
We demonstrate that the AEA results are comparable to those obtained with an accurate head model.