Hi, Julian! Well, what happens is that an adaptive beamformer (AB) in fact solves optimally estimation problem in a space of fixed number of active sources ( whose topographies enter G_i enter the constraints equations(CE) W'T*G_i = k_i, i=1,N, N - constraints count). To do so, it basically, first spatially whitens the data and then estimates timeseries by projecting the whitened data onto the topographies(transformed into the whitened space as well) of all the N constraints(zero and non-zero) to estimate the timeseries of the sources whose topographies are used in the CEs. Then, it sums these estimated timeseries scaling them by k_i and produces the output. As long as the number of constraints N < than the # of channels you will always be able to do the exact inverse and guarantee exactly the specified gain factors k_i at the directions G_i and thus satisfy the constraints equations.

Well, the question now is what is actually happening to those sources not specified in the CEs. The answer is that due to whitening their power is getting uniformly spread out over the entire space and therefore minimum possible amount of these not-specified noise sources leaks into the estimates of activity of all of the sources mentioned in the CEs. It leaks in all of them, but uniformly because the whitening reduces the problem of estimation to that of estimation in the white noise case. So, when jargon is used that the AB "puts zero into the direction of the most powerful sources" it is not exactly so, not exactly zero, unless this most powerful source topography enters the CEs with k_i = 0. Apparently, you can not specify all of the noise sources (for one thing you don't know them, and for another is that then you are likely to run out of degrees of freedom(depends of the point of viewon the brain activity and on the time-window width) and will arrive again to the underdetermined case of the classical min-norm )

Hope this helps,

Alex

HI Julien,

in a nutshell, the famous 'unit gain' constraint in the beamformer equations guarantees that if you want to reconstruct the signal from a source x(r1,t) with beamformer coeffcients vector w(r1) from the MEG measurements M(t), you always get:

estimate(x(t)) = w(r1)*M(t)=x(r1,t) + rest

Especially if there is onlyx a single active source:

estimate(x(t)) = w(r1)*M(t)=x(r1,t)

This means, the correct time course is always there, albeit potentially hidden by 'the rest'. The minimization of the rest is possible now because the unit gain constraint does not determine the weight vector w(r1) completely, and changing this weight vector while sticking to this unit gain constraint allows you to change 'the rest'.

In principle you could scan all weight vectors that are allowed by the unit gain constraint until you get a minimal rest. Since you do not know how your estimate decomposes into x(r1,t) and rest, you just minimize the power of your estimate. Because of the unit gain constraint that you respect with the weights w(r1) that you try, this will 'never' take away signal from the x(r1,t) part, only from the rest - so minimzing power of the total estimate is in principle sufficient to minimize interference.

Notes:

I put the 'never' in quotation marks, because if the x(r1,t) and the rest are correlated you can actually find weights that are allowed under the unit gain constraint but weight the rest negative such that it cancels x(r1,t), getting even lower power values of the estimate than for the correct solution - this is where the famous problem with the highly correlated sources and the beamformer comes from.

Instead of first writing down all possible weights allowed under the unit gain constraint and then searching for the one with the minimal power of the estimate 'brute force', you can of course use a closed form solution from linear algebra. This is what is done - it's just not as easy to see in these formulas how the minimization of the interference is done. I hope my description made the mechanics of this minimization a bit more clear.

I have a bookchapter on source analysis in Debener and Ullsperger 'simultaneous eeg and fmri' that basically repeats the above description in some more detail.

I hope this helps, please don't hesitate to ask for further clarification.

Michael

I forgot to say that a minimal interference does not necessarily mean a small interference. Even after minimization, the 'rest' noise amplitude may be an order of magnitude bigger than the signal that you want, which is why you typically contrast two beamformer images, with the same noise but differences in the activity of interest.

Julian,

Michael provided you with a great intuitive answer, and my answer complements it by providing some additional intuition in case you have estimation theory in your bckg . For more formal treatments, please consult K Sekihara and B. Van Veen papers. Also, some while ago we had a paper where we were treating the BFs very formally and exploring their properties. Greenblatt et all, IEEE paper. Keep in mind that additional problem of adaptive BFs is that of signal cancellation that happens when the model used to form the constraints is not accurate enough. ironically, in this case BFs perform better with the data with low channel count as under low channel count scenario they don't have enough degrees of freedom to completely cancel the target signal.. :) This is especially the case when using ADAPTIVE BFs in analysis of EEG data where high forward model accuracy may be hard to guarantee.

Thanks a lot! This really helped.

Dear Julian,

it is difficult to explain the minization procedure within a few lines, but have a look at the following (open access) book

http://www.intechopen.com/books/magnetoencephalography

In particular chapter 4 nicely leeds you through the beamformer technique including the different minimization procedures used in the community.

http://www.intechopen.com/books/magnetoencephalography/characterization-of-brain-dynamics-using-beamformer-techniques-advantages-and-limitations