# Interference Suppression in EEG Dipole Source Localization through Reduced-Rank Beamforming
## 1 Introduction  
Spatial filtering techniques (also known as _beamformers_) have been used in electroencephalography (EEG) and magnetoencephalography (MEG) for signal reconstruction and localization of sources of brain electrical activity. In most cases, the proposed methods are based on fitting dipole sources within a region-of-interest (ROI) which is chosen based on anatomical restrictions and assumptions about the relationships between brain sources [1]. For example, the linearly-constrained minimum variance beamformer (LCMV) introduced in [2], has been proved to be effective in localizing sources of brain electrical activity from surface recordings under the consideration that such sources have minimum correlation [3].  
LCMV-based solutions and, to some extent, all source activity estimation methods based on beamforming underperform if the EEG forward model becomes ill-conditioned due to sources being closely positioned, possibly correlated, in presence of interference, or with poorly estimated signal and noise covariance matrices [3; 4; 5]. In order to overcome that, different reduced-rank estimators have been proposed, as they offer a gain in performance compared to the LCMV-solutions for estimating dipole source signals using EEG/MEG in ill-conditioned settings [6; 7]. Most recently, in [8], a minimum-variance pseudo-unbiased reduced-rank (MV-PURE) estimation framework is proposed, which provides higher spatial resolution than LCMV-based solutions. However, the selection of the level of the rank-reduction is still an issue related to these methods.  
The general consensus is that not a single source localization method is capable of providing high sensitivity and specificity at the same time, hence overlapping approaches might particularly be of help in the identification of sources whose activity may be relevant in specific scenarios [9]. In particular, we are interested in studying filters that arespecialized in reducing the influence of interfering sources, as they may be useful in applications where it is important to have such interferences removed from the reconstructed activity, e.g., in applications of directed connectivity measures such as partial directed coherence [10]. Hence, previous efforts of our group and collaborators have focused on reduced-rank beamformers that explicitly consider brain activity originating in regions different to those of main interest, but measured as correlated with signals of interest by the EEG sensor array (see [11] and references therein). The reduced-rank approach provides a significant gain in performance in such settings, as it introduces a small amount of bias in exchange for large savings in variance [12]. Our group have also previously proposed a technique in which additional restrictions on the ROIs are considered based on the _sparsity_ of the neural activity [13], and such an approach is inspired in the rank-reduction provided by the _eigencancel_[7].  
Adding up to the listed efforts, in this paper we present an approach based on the structure of the generalized sidelobe canceler (GSC) [14]. The proposed source localization methods are based on the optimization of _neural indices_ that are designed to reduce the contribution of the noise and interference components to the output of the GSC. This idea was initially explored in [15] but in its _quiescent_ (non-adaptive) form, which is known to be LCMV-optimal only if the interference signals arrive from specific directions (see [16] for details). Instead, we propose the use of a modified structure in which signal components are adaptively selected from the noise subspace of the GSC in order to select the optimal rank-reduction required for effective interference suppression [17].