1. Beamformer dimensionality ScalarVector Features1 optimal source orientation selected per location. Wrong orientation choice may lead to missed sources. 2 (tangential) or 3 orthogonal source orientations per location, power summed. 2 or 3 times as much projected noise power compared to scalar with correct orientation. Source power Beamformer weights (units depend on constraints) Source orientation of maximal power Constraint type Minimization problem constraints WeightsFeatures Minimization problem constraints WeightsFeatures Unit-gain or distortionless, power in (Am) 2 (Similarities with corresponding scalar solutions, but there are bias differences. Needs closer inspection.) Array-gainNo location bias. Absolute values not very meaningful; indicates the relative sensor array sensitivity (or gain). Unit-noise-gain, normalized weights, power = SNR + 1 (pseudo-Z) “Scalar-like” Normalize array-gain solution References Linearly constrained minimum variance beamformer http://dx.doi.org/10.6084/m9.figshare.1148970 Rotational invariance and source orientation in LCMV vector beamformer Marc Lalancette marc.lalancette@sickkids.ca Diagnostic Imaging, The Hospital for Sick Children, Toronto, Ontario, Canada IntroductionProposed new constraints and weights (highlighted), and results Rotational invariance Adaptive spatial filter commonly used to solve the inverse problem in MEG, i.e. to determine the neuronal activity at the source of the detected magnetic field surrounding the head. Finds an optimal linear combination of the sensors (weights vector) to represent each possible source location and orientation, minimizing the power from other locations and orientations while maintaining the desired source amplitude by an appropriate choice of linear constraints. Any physically meaningful quantity must be independent of the choice of coordinate system orientation. Such rotationally invariant quantities can always be represented as tensor equations (e.g. composed of vectors and matrices), without any individual vector components or matrix elements. Examples of methods that are not rotationally invariant: Normalizing columns of a matrix, such as the lead field matrix in the array-gain vector constraint 1 ; Commonly used unit-noise-gain vector constraint 1 ; Adding scalar solutions in orthogonal directions instead of using a vector formulation 2,3. These can be seen as adding distortions in the reconstructed activity that can affect amplitude and localization. The strength of these effects remains to be investigated, but could lead to missed sources in specific cases. 1 Sekihara K and Nagarajan S S, Adaptive Spatial Filters for Electromagnetic Brain Imaging, 2008, Springer-Verlag, Berlin. 2 Huang M-X et al., Commonalities and Differences Among Vectorized Beamformers in Electromagnetic Source Imaging, Brain Topography, 16:3, 2004, p 139-58 3 Johnson S et al., Examining the Effects of One- and Three-Dimensional Spatial Filtering Analyses in Magnetoencephalography, 2011, PLoS ONE 6(8): e22251. Conclusion The new vector beamformer constraints and solutions presented here are rotationally invariant and thus avoid an unnecessary source of potential amplitude and localization distortions. Preliminary results indicate they are otherwise comparable to or better than previous similar scalar and vector solutions in terms of orientation bias and spatial resolution, as expected. Further investigation should clarify which types are optimal in typical situations. We present here new rotationally invariant constraints for the vector beamformer and compare the resulting solutions with other vector and scalar options in terms of location and orientation bias and resolution. Distributions of various system and source parameters used to chose typical values for bias figures on the right.