= Beamforming methods =
''Authors: Hui-Ling Chan''

The estimation of source distribtion is an important step to understand the brain activity from EEG and MEG data. Dipole fitting, minimum norm estimation, and beamformer are three commonly used methods. It has been proved that beamforming methods provide good spatial resolution. This tutorial will show how to apply beamforming methods to MEG data and obtain the statistic map of source activation.

We are going to use the protocol '''!TutorialRaw''' created in the previous tutorial [[Tutorials/TutRawAvg|Epoching and Averaging]]. If you have not followed this tutorial yet, please do it now.

<<TableOfContents(2,2)>>

== Introduction ==
Beamfoming methods scan each targeted voxel/vertex position <<latex($\bf r$)>> and estimate the spatial filter <<latex(${\bf w}_{\bf r,q}$)>>. By multiplying with the MEG recordings <<latex(${\bf m}(t)$)>>, the spatial filter <<latex(${\bf w}_{\bf r,q}$)>> outputs the temporal waveform <<latex($y_{\bf r,q}(t)$)>>  of the dipole source at that position with the dipole orientation <<latex($\bf q$)>> as below:

<<latex(\begin{equation} y_{\bf r,q}(t) = {{\bf w}_{\bf r,q}}^{\rm T}{\bf m}(t)\enspace,  \end{equation})>>

where 'T' indicates the transpose of a matrix or vector .

==== Vector-type beamformer ====
For each position <<latex($\mathbf{r}$)>>, three orthogonal spatial filters <<latex($\mathbf{W}_{\mathbf{r}}=\left[\mathbf{w}_{\mathbf{r},\mathbf{q}_x},\mathbf{w}_{\mathbf{r},\mathbf{q}_y},\mathbf{w}_{\mathbf{r},\mathbf{q}_z}\right]$)>> are computed by applying the unit-gain constraint as well as the minimum norm and minimum variance criteria as below:

<<latex(\begin{equation} \hat{\mathbf{W}}_{\mathbf r}=(\mathbf{C}_{\rm s}+\alpha\mathbf{I})^{-1}\mathbf{L}_\mathbf{r}\left(\mathbf{L}_\mathbf{r}^{\rm T}(\mathbf{C}_{\rm s}+\alpha\mathbf{I})^{-1}\mathbf{L}_\mathbf{r}\right)^{-1} \enspace, \end{equation})>>

where <<latex({\bf C}_{\rm s})>> is the covariance matrix of MEG recordings during window <<latex($T_{\rm_s}$>>, <<latex($\mathbf{I}$)>> is the identity matrix, <<\latex($\mathbf{L}_{\mathbf{r}}$)>> is the gain matrix for the dipole located at position <<latex($\mathbf{r}$)>>, and <<latex(\alpha)>> is the regularization parameter which compromises the minimum norm and minimum variance criteria.

==== Scalar-type beamformer ====
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== Linearly-constrained minimum variance beamformer (LCMV) ==
==== Section 1 ====
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==== Section 2 ====
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== Maximum constrast beamformer (MCB) ==
==== Section 1 ====
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==== Section 2 ====
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== Beamformer-based correlation/coherence imaging ==
==== Dynamic imaging of coherent sources (DICS) ====
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==== Spatiotemporal imaging of linearly-related source components (SILSC) ====
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