Hello Rik,

The total noise covariance consists of brain and sensor noise, and SSS 
only modifies the latter because the former is produced by the brain and 
thus faithfully represented by the SSS basis. It is important to 
remember that the brain noise dominates the noise covariance of MEG 
signals at relatively low frequencies that typically contain most of the 
interesting brain phenomena. Thus, the changes in sensor noise 
covariance due to application of MaxFilter have not been considered 
significant enough to be worth taking into account in practice.

Equation 34 of the 2005 JAP paper describes the covariance of the 
SSS-processed data as a combination of the true signal covariance 
(C_phi) and the modified sensor noise, which is the latter part of the 
formula. Matrix N represents the original sensor noise covariance, 
typically assumed diagonal, and the modified sensor noise covariance can 
then be analytically calculated by eq. 34 in the form 
S_in*P*pinv(S)*N*pinv(S)'*P'*S_in'. If desired, this contribution could 
be subtracted from the measured covariance. A practical way to estimate 
the sensor noise covariance could be accomplished, e.g., by simulating a 
file with artificially generated random noise, applying MaxFilter to it, 
and finally calculating the covariance from the resulting file.

Best regards,
Samu


Rik Henson wrote:
> Samu -
>
> Many thanks for your detailed reply, which clarifies a lot.
>
> Could I beg your time on one further issue: As Danny Mitchell pointed 
> out to me, in your thesis you state:
>
> "In MEG measurements, it is usually assumed that this noise is 
> normally distributed and uncorrelated among the sensors, resulting in 
> a diagonal covariance matrix. Application of SSS changes the sensor 
> noise covariance, which can be taken into account if needed as shown 
> in publication II [Taulu 2005 JAP]."
>
> I looked at your 2005 JAP paper, but still need some help as to how in 
> practice to extract/estimate the changes in sensor noise covariance 
> resulting from applying MaxFilter?
>
> Thanks
> Rik
>
>
> Samu Taulu wrote:
>> Dear Colleagues,
>>
>> This is a very interesting discussion. I hope I can clarify some of 
>> the questions in the previous emails.
>>
>> First of all, I would like to briefly describe the way how SSS 
>> affects the data in general. Apparently, the number of degrees of 
>> freedom of an MEG signal is very high, in the case of Elekta Neuromag 
>> Oy the rank of the data is 306 to begin with. However, due to the 
>> sampling theory of neuromagnetic fields and the relatively low signal 
>> to noise ratio, most of the dimensions in the signal space belong to 
>> random sensor noise while the dimensionality of the brain and 
>> external interference signals is much smaller. This means that only 
>> around 100 field components are needed to practically represent the 
>> signals of interest and interference. The SSS method has been 
>> designed in such a way that it models those basic components with 
>> vector spherical harmonic expansions that are truncated at the limit 
>> above which the components fall under the sensor noise level. In 
>> accordance with the sampling theory, these components correspond to 
>> very high spatial frequencies of the magnetic field. The internal 
>> (brain) and external (interference) signals are both included in the 
>> SSS matrix, and generally the angle between an arbitrarily chosen 
>> pair of internal and external basis vectors is less than 90 degrees, 
>> i.e., there is overlapping between them. The SSS basis, however, is 
>> linearly independent and the decomposition into those basis 
>> components is unique. Assuming that the sampling theory and 
>> quasistatic Maxwell's equations hold, the brain signal estimate does 
>> not leak into the external part and thus the spatial overlapping of 
>> the internal and external signals does not cause a need for leadfield 
>> corrections, unlike in the case of SSP where the orthogonal spatial 
>> projection slightly modifies the brain signals. The effect of SSP is 
>> compensated for in the Xfit software, for example.
>>
>> The potential distortion to brain signals caused by SSS would happen 
>> due to the truncation of the vector spherical harmonic expansion. We 
>> have examined the effect of the truncation by simulations in Figs. 
>> 1-4 of our paper (Taulu S, Simola J, Kajola M, IEEE Trans. Sign. 
>> Proc., vol. 53, pp. 3359-3372 (2005)) and found out that the effect 
>> is practically insignificant. In other words, manipulation of the 
>> leadfields should not be necessary after SSS. If you like, you can 
>> create the transformation matrix like Olaf suggested - MaxFilter does 
>> not return such a transformation.
>>
>> If you would like to experiment the effect of SSS on the leadfields, 
>> you could try the following simple experiment:
>> 1. Simulate the signal of any reasonable current dipole in Xfit. This 
>> step utilizes full-rank leadfields with no linear transformation 
>> performed on the data.
>> 2. Run MaxFilter on the simulated file
>> 3. Load the output file of MaxFilter into Xfit and perform source 
>> analysis
>>
>> In step 3, Xfit assumes original leadfields without any matrix 
>> manipulation and therefore the possible discrepancy is directly 
>> assessed by comparing the results obtained with the original and 
>> SSS-processed data because Xfit treats both of them in the same way: 
>> Without leadfield correction. Based on our theory and experiments, 
>> there should be no significant distortion in the field pattern or 
>> source localization even without any leadfield manipulations.
>>
>> I hope this clarifies the issue, and sorry for the length of this email.
>>
>> Best regards,
>> Samu
>>
>>
>> Olaf Hauk wrote:
>>> If there is no "built in" way: One could create an artifical data 
>>> set that
>>> contains only the identity matrix (n*n, n: number of sensors), and 
>>> apply
>>> SSS to that in order to get the transformation matrix.
>>>
>>> Olaf
>>>
>>>
>>>
>>>  
>>>> I think this is the key, Olaf.
>>>>
>>>> I apologise that my original email caused some confusion, because I 
>>>> was
>>>> not asking specifically about the temporal extension of SSS, but 
>>>> rather
>>>> the use of SSS generally.
>>>>
>>>> I would also hope that the SSS components reflecting environmental 
>>>> noise
>>>> sources in the outer sphere are only a small part of the sensor space
>>>> spanned by the leadfield matrix, so their removal would have little
>>>> affect on that matrix.
>>>>
>>>> However, my question remains: in order to compare "with and without"
>>>> leadfield correction (as Olaf suggests), how do I extract the 
>>>> necessary
>>>> correction (projection) from MaxFilter?
>>>>
>>>> Advice from MaxFilter experts (ie Neuromag?) much appreciated....
>>>>
>>>> Rik
>>>>
>>>>
>>>>   
>>>>> Hi again, and sorry: I was a bit too quick, there is of course no 
>>>>> "null
>>>>> space" in sensor space of the leadfield. But the question how much 
>>>>> the
>>>>> SSS
>>>>> components have in common with the leadfield still remains, and if 
>>>>> the
>>>>> overlap is small, how big the effect of such a leadfield 
>>>>> correction on
>>>>> source estimates really is. Has anyone tried with and without?
>>>>>
>>>>> Olaf
>>>>>
>>>>>
>>>>>
>>>>>     
>>>>>> I would say one has to think about SSSt as a temporal filtering 
>>>>>> method
>>>>>> -
>>>>>> and you don't correct your leadfield after low-pass filtering 
>>>>>> either,
>>>>>> for
>>>>>> example. But this raises another interesting question: If SSS 
>>>>>> (with or
>>>>>> without ST) only removes activity from sources outside the sensor
>>>>>> array,
>>>>>> it should only remove patterns that are in the null space of the
>>>>>> leadfield
>>>>>> - i.e. no correction would be required. If it removes patterns 
>>>>>> that are
>>>>>> NOT in the null space of the leadfield, these sources could 
>>>>>> potentially
>>>>>> be
>>>>>> generated inside the head (where the brain is) - i.e. it might 
>>>>>> remove
>>>>>> signal! I would hope that it's the former.
>>>>>>
>>>>>> Olaf
>>>>>>
>>>>>>
>>>>>>
>>>>>>       
>>>>>>> Burkhard -
>>>>>>>
>>>>>>> I thought so too, but another colleague thought this was not the 
>>>>>>> case.
>>>>>>> So if the Neuromag experts don't give the definite answer, we could
>>>>>>> have
>>>>>>> a vote?
>>>>>>>
>>>>>>> ;-)
>>>>>>> Rik
>>>>>>>
>>>>>>> Burkhard Maess wrote:
>>>>>>>
>>>>>>>         
>>>>>>>> Hi Rik,
>>>>>>>>
>>>>>>>> this is an interesting question - but I think the temporal 
>>>>>>>> projection
>>>>>>>> does not modify the spatially organized leadfield. SSSt takes 
>>>>>>>> out the
>>>>>>>> part of the data which correllates highly between both expansions,
>>>>>>>> but
>>>>>>>> you can not describe it by a certain spatial pattern as in the 
>>>>>>>> case
>>>>>>>> of
>>>>>>>> the SSP.
>>>>>>>>
>>>>>>>> Cheers,
>>>>>>>> Burkhard
>>>>>>>>
>>>>>>>>
>>>>>>>> Rik Henson wrote:
>>>>>>>>
>>>>>>>>           
>>>>>>>>> Dear Neuromeg -
>>>>>>>>>
>>>>>>>>> Could you let me know how I can correct my leadfields for prior
>>>>>>>>> SSSt?
>>>>>>>>> In other words, I have a leadfield matrix, L, of n sensors x p
>>>>>>>>> sources, and would like to extract some form of projection matrix
>>>>>>>>> from Maxfilter that I can apply to L in order to remove those
>>>>>>>>> components of the sensor data that have been removed by SSSt.
>>>>>>>>>
>>>>>>>>> Many thanks
>>>>>>>>> Rik
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>               
>>>>>>> -- 
>>>>>>>
>>>>>>> -------------------------------------------------------
>>>>>>>                  Dr Richard Henson
>>>>>>>          MRC Cognition & Brain Sciences Unit
>>>>>>>                  15 Chaucer Road
>>>>>>>                    Cambridge
>>>>>>>                   CB2 7EF, UK
>>>>>>>
>>>>>>>            Office: +44 (0)1223 355 294 x522
>>>>>>>               Mob: +44 (0)794 1377 345
>>>>>>>               Fax: +44 (0)1223 359 062
>>>>>>>
>>>>>>> http://www.mrc-cbu.cam.ac.uk/people/rik.henson/personal
>>>>>>> -------------------------------------------------------
>>>>>>>
>>>>>>>
>>>>>>>           
>>>>>> -- 
>>>>>>
>>>>>>
>>>>>>         
>>>>>
>>>>>       
>>>> -- 
>>>>
>>>> -------------------------------------------------------
>>>>
>>>> DR RICHARD HENSON
>>>>
>>>> MRC Cognition & Brain Sciences Unit
>>>> 15 Chaucer Road
>>>> Cambridge, CB2 7EF
>>>> England
>>>>
>>>> EMAIL:  [log in to unmask]
>>>> URL:    http://www.mrc-cbu.cam.ac.uk/people/rik.henson/personal
>>>>
>>>> TEL     +44 (0)1223 355 294 x522
>>>> FAX     +44 (0)1223 359 062
>>>> MOB     +44 (0)794 1377 345
>>>>
>>>> -------------------------------------------------------
>>>>
>>>>     
>>>
>>>
>>>   
>>
>>
>