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| Beamfoming methods scan each target voxel/vertex position '''r''' to estimate a spatial filter <<latex({\bf W}_{\bf r})>> which outputs the source activity y | 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: |
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| ==== Spatial filter ==== Text |
<<latex(\begin{equation} y_{\bf r,q}(t) = {{\bf w}_{\bf r,q}}^{\rm T}{\bf m}(t)\enspace, \end{equation})>> |
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| ==== Section 2 ==== | 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 ==== |
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 ?Epoching and Averaging. If you have not followed this tutorial yet, please do it now.
Contents
Introduction
Beamfoming methods scan each targeted voxel/vertex position
latex error! exitcode was 2 (signal 0), transscript follows:and estimate the spatial filter
latex error! exitcode was 2 (signal 0), transscript follows:. By multiplying with the MEG recordings
latex error! exitcode was 2 (signal 0), transscript follows:, the spatial filter
latex error! exitcode was 2 (signal 0), transscript follows:outputs the temporal waveform
latex error! exitcode was 2 (signal 0), transscript follows:of the dipole source at that position with the dipole orientation
latex error! exitcode was 2 (signal 0), transscript follows:as below:
latex error! exitcode was 2 (signal 0), transscript follows:
where 'T' indicates the transpose of a matrix or vector .
Vector-type beamformer
For each position
latex error! exitcode was 2 (signal 0), transscript follows:, three orthogonal spatial filters
latex error! exitcode was 2 (signal 0), transscript follows:are computed by applying the unit-gain constraint as well as the minimum norm and minimum variance criteria as below:
latex error! exitcode was 2 (signal 0), transscript follows:
where
latex error! exitcode was 2 (signal 0), transscript follows:is the covariance matrix of MEG recordings during window
latex error! exitcode was 2 (signal 0), transscript follows:is the identity matrix, <<\latex($\mathbf{L}_{\mathbf{r}}$)>> is the gain matrix for the dipole located at position
latex error! exitcode was 2 (signal 0), transscript follows:, and
latex error! exitcode was 2 (signal 0), transscript follows:is the regularization parameter which compromises the minimum norm and minimum variance criteria.
Scalar-type beamformer
Text
Linearly-constrained minimum variance beamformer (LCMV)
Section 1
Text
Section 2
Text
Maximum constrast beamformer (MCB)
Section 1
Text
Section 2
Text
Beamformer-based correlation/coherence imaging
Dynamic imaging of coherent sources (DICS)
Text
Spatiotemporal imaging of linearly-related source components (SILSC)
Text