Dear Octavian,
Good question, like usual. Yes, the resolution matrix will be:
R = Kernel.Imaging * Headmodel.Gain;
This matrix maps the space of ‘true’ currents to the space of estimated currents.
The point spread functions will be its column vectors. The ith column (i.e., psf) will be a map of how true activity in the ith dipole component spreads from that source point to all over the source space during estimation.
The resolution kernels will be the row vectors. These will be images of how activity from all over the source space contributes to the ith component estimate.
Now, keep in mind that in BST, we usually recommend for people to whitened the data. And if the data has been processed with SSS, SSP, or any other dimensionality reduction denoising algorithm, the noise covariance will not be full rank.
That means that the whitening operator will not be square. It will have more columns than rows. Thus, the whitened data will have less signals than the original data. This will also be true about the whitened lead field matrix. And it will also imply that the inverse operator will technically have less column vectors than there are sensors. BUT, in BST, we have taken the convention of pre-multiplying the inverse operator from the right side with the whitening operator. This is why the ImagingKernel matrix will have as many columns as sensors. Thus, when this imaging kernel is multiplied with the unwhitened data, it automatically whitens the data, and estimates the sLORETA activity. This will also be true of when you multiply the imaging kernel with the unwhitened lead field matrix, thereby whitening it.
I wrote this function to compute the sLORETA estimate to be equivalent to the Pascual-Marqui paper. But I should make a few comments to clarify details. First of all, we are using whitening to model and suppress correlated noise. Also, we are computing the MNE part of the sLORETA the same way that the MNE Suite does it. This means that the SNR2 can be modeled as the ratio of the whitened data power, taken into account the fact that a dimensionality reduction may have occurred. This implies that the regularization parameter lambda2=1/SNR2. Also, it implies that the source covariance matrix has been adjusted as a function of the trace of LC_sL’, so that the trace is equal to the rank of the noise covariance, where L is the lead field matrix and C_s is the source covariance, i.e., a multiple of the identity matrix. I recommend checking out the MNE Suite Manual for more details. Also, check at the matlab code. It’s well commented.
Also, I should mention that all the sLOERETA computations related to the inverse block resolution matrices have been taken care of in a for loop in such a way that when you multiply your data or your lead field matrix with the ImagingKernel, it is directly producing the sLORETA activity. In the case of fixed orientation constraints, taking the absolute value of the activity will give you Pascual-Marqui’s absolute standardized activity. But you don’t have to take that absolute value if you want to for example look at non-rectified oscillations in the source domain. In the case, of free dipole orientations, then you have 3 components to the sLORETA activity per source point. If you square them and sum them, you have the sLORETA power, and if you further take the square root you have a modulus of sLORETA activity. These will be rectified, so don’t do frequency analysis at this step. Do it at the sensor level, or at the sLORETA component level, before taking the power.
In BST, we allow sLORETA solutions to have loose orientation constraints, in addition to fixed and free. Personally, I would recommend to use free or loose orientation constraints. In both cases, you will have three components per source point. For statistical contrasts, the ERFs can be subtracted so that the sLORETA reflects a difference, for example, or likewise you can subtract the sLORETA activity components from the two conditions. I believe that in BST the effective number of averages is taken care of by dividing the noise covariance appropriately, taking into account the number of trials in the two conditions. If not, or to be sure, I would take the difference at the source level of sLORETA activity per component, and then take the modulus of the difference sLORETA map, before moving the group analyses. Contrasts in the sensor or source domain should give the same exact results, literally exactly the same. Of course, there are other statistical hypotheses that can be made, actually many many versions, and it all depends on the questions being asked, e.g., statistical differences within subject averaging across epochs, versus group statistics relative to pre-stimulus baseline, or relative to a different conditions, etc, etc, etc. It gets complicated. Just mentioning a few cases for people to be aware of some of the possibilities.
Please don’t hesitate to ask more questions, and I will try to answer as best as possible.
Cheers,
Rey