# Interference Suppression in EEG Dipole Source Localization through Reduced-Rank Beamforming
## 2 Materials and Methods
### Beamforming and Neural Indices  
Based on the measurements in (2), a beamformer \(W\) can be proposed in such a way that the dipole's signals could be approximated from the measured data, i.e., \(\widetilde{Q}_{k}=W^{T}Y_{k}\). Then, a consistent estimate of \(Q\) could be obtained as \(\widetilde{Q}=\sum_{k}^{K}\widetilde{Q}_{k}/K\) for a sufficiently large value of \(K\). Furthermore, the beamformer is usually designed in order to satisfy the following condition:  
\[W^{T}(\mathbf{r})A(\mathbf{r})=\left\{\begin{array}{ll}I&\text{if}\quad\mathbf{r}=\mathbf{r} _{l}\\ \mathbf{0}&\text{if}\quad\mathbf{r}\
eq\mathbf{r}_{l}\end{array}\right.. \tag{4}\]  
In the context of brain signals, the conditions in (4) enforce that \(W\) is designed to pass activity coming from a specific location of interest \(\mathbf{r}_{l}\), while it attenuates signals coming from other locations \(\mathbf{r}\), for \(\mathbf{r}\in\Omega\), where \(\Omega\) denotes the volume of the brain.  
There are many ways to design a beamformer, but the LCMV approach to achieve optimality is quite effective: to minimize the variance at the filter's output while satisfying the linear response constraint \(W^{T}(\mathbf{r}_{l})A(\mathbf{r}_{l})=I\). The solution to such minimization problem may be obtained using Lagrange multipliers (which is the classical method for finding local minima of a function subject to equality constraints) and completing the square, which results in [20]  
\[W(\mathbf{r}_{l})_{\text{LCMV}}=\left[A^{T}(\mathbf{r}_{l})R^{-1}A(\mathbf{r}_{l})\right]^ {-1}A^{T}(\mathbf{r}_{l})R^{-1}, \tag{5}\]where \(R=E[YY^{T}]\) corresponds to the data's covariance matrix. For the case of unknown \(R\), a consistent estimate of this covariance matrix (denoted by \(\widehat{R}\)) can be used.  
Furthermore, high variability in the output of the LCMV beamformer as a function of position can be considered indicative of significant neuronal activity [3]. Therefore, to find a source through this approach requires an exhaustive search for the maximum value of the _neural activity index_ within \(\Omega\). This is written as  
\[\hat{p}_{l}=\max_{\mathbf{r}}\ \frac{\operatorname{tr}\left\{\left[A_{l}^{T}(\mathbf{r}_ {l})\widehat{R}^{-1}A(\mathbf{r}_{l})\right]^{-1}\right\}}{\operatorname{tr} \left\{\left[A_{l}^{T}(\mathbf{r}_{l})\widehat{P}^{-1}A(\mathbf{r}_{l})\right]^{-1} \right\}}=\max_{\mathbf{r}}\ \iota_{\text{LCMV}}(\mathbf{r}), \tag{6}\]  
where \(\iota_{\text{LCMV}}(\mathbf{r})\), denotes the neural activity index, \(\widehat{P}\) is an estimate of the noise covariance matrix, and \(\operatorname{tr}\{\cdot\}\) indicates the trace. Note that (6) is equivalent to maximize the source's variance (normalized by the variance of the noise) as a function of \(\mathbf{r}\).  
Since the LCMV filter can only be applied for one source at a time, then the multi-source activity index (MAI) is proposed in [21] as an extension of \(\iota_{\text{LCMV}}(\mathbf{r})\) for the case of \(L\) neural sources:  
\[\iota_{\text{MAI}}(\mathbf{\theta})=\operatorname{tr}\left\{G(\mathbf{\theta})H(\mathbf{ \theta})^{-1}\right\}-3L, \tag{7}\]  
where  
\[G(\mathbf{\theta})=A^{T}(\mathbf{\theta})\widehat{P}^{-1}A(\mathbf{\theta}), \tag{8}\]  
and  
\[H(\mathbf{\theta})=A^{T}(\mathbf{\theta})\widehat{R}^{-1}A(\mathbf{\theta}). \tag{9}\]  
Theoretically, maximizing (7) provides an unbiased estimation of \(\mathbf{\theta}\) for any signal-to-noise ratio and any level of correlation between the sources. Yet, in practice, these factors strongly affect the spatial resolution of \(\iota_{\text{MAI}}\).