Reconstruction of the neuronal current inside the human brain from non-invasive measurements of the magnetic flux density via magnetoencephalography (MEG) or of electric potential differences via electroencephalography (EEG) is an invaluable tool for neuroscientific research, as it provides measures of activity in the brain. However, it is also a severely ill-posed inverse problem. Assuming spherical geometries, we consider the spherical multiple-shell model for the inverse MEG and EEG problem and apply the regularized functional matching pursuit algorithm (RFMP) for its solution. We present a new convergence proof for the RFMP for operators between two infinite-dimensional Hilbert spaces. Moreover, we utilize the complementarity of EEG and MEG data to combine inversions of simultaneous electric and magnetic measurements. Finally, we test the algorithm numerically on synthetic data using several Sobolev norms as penalty term and apply it to real data.
Keywords: electroencephalography, greedy algorithms, ill-posed problems, integral equation, inverse problems, magnetoencephalography, regularization methods, regularized functional matching pursuit, Sobolev spaces
