Beamformer dimensionality ScalarVector Features1 optimal source orientation selected per location. Wrong orientation choice may lead to missed sources.

Beamformer dimensionality Scalar Vector

Features 1 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 powerBeamformer weights(units depend on constraints)

, subject to , all collinear along . , subject to

the matrices on the right maintain the full subspace defined by the columns of .

Source orientation of maximal power

Constraint typeMinimization problem

constraints Weights Features Minimization problem constraints Weights Features

Unit-gain or distortionless, power in (Am)2

Biased towards regions of smaller (middle of the MEG sensor array, or center of a spherical head model) and towards radial orientations.

(Similarities with corresponding scalar

solutions, but there are bias differences. Needs

closer inspection.)

Array-gain or No location bias. Absolute values not very meaningful; indicates the relative sensor array sensitivity (or gain).

or

Unit-noise-gain, normalized weights, power = SNR + 1 (pseudo-Z)

Impose on unit- or array-gain solution: Effectively:

No location bias, better resolution. Amplitude depends on how far is rotated away from . Once the unbiased location is determined, the unit-gain power at that location should be favored for amplitude comparisons.

“Scalar-like”Normalize () unit-gain solution: weight subspace projection matrix

Normalize array-gain solution orthonormalized

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 [email protected]

Diagnostic Imaging, The Hospital for Sick Children, Toronto, Ontario, Canada

Introduction Proposed 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 constraint1;• Commonly used unit-noise-gain vector constraint1;• Adding scalar solutions in orthogonal directions instead of

using a vector formulation2,3. 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.

1Sekihara K and Nagarajan S S, Adaptive Spatial Filters for Electromagnetic Brain Imaging, 2008, Springer-Verlag, Berlin.2Huang M-X et al., Commonalities and Differences Among Vectorized Beamformers in Electromagnetic Source Imaging, Brain Topography, 16:3, 2004, p 139-583Johnson 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.

Rotational variance in array-gain and unit-noise-gain vector solutions. As the x and y axes are rotated tangentially, the total projected amplitude (dashed lines) changes. Solid lines: x-component contribution: . (See bias figure description box for additional details.)

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.

Source bias and resolution figures: All values are normalized by the simulated source field amplitude and for unit-gain. is the angle from the simulated source orientation to the physical orientation corresponding to the weights, rotating in the “tangential source” plane. Dashed lines: square root of total projected power (trace for vector solutions – right, rotationally variant sum of components and for scalar – left). Solid lines: contribution of the first component, either or . Input signal to noise ratio (SNR) 0.85 unless specified otherwise. Dot-dashed lines (last figure only): non rotationally-invariant solutions.

, uniform noise, uniform noise , uniform noise, SNR=50 , uniform noise, SNR=50

, uniform noise , uniform noise, full noise covariance , full noise covariance

, 2 sources, uniform noise , 2 sources, uniform noise, uniform noise, SNR=100 , uniform noise , SNR=100