Matt,
Unfortunately, it's not quite the same (mathematically).
The log-signal is quadratic in the bvecs, rather than linear, so the two are not equivalent.
Another way to see this is: imagine you have a fibre direction x, and two bvecs that have the same angle with x (say a dot product of 0.9), but the average of the two bvecs is exactly x (+ and - the same angle from x for example).
In such a case, the prediction from the average bvecs will be different from the average prediction.
This effect is anisotropic: it will be worse along x than across x. This is due to the extra terms in the quadratic form that are proportional to the dot product between bvecs and x.
Cheers,
Saad.
On 22 Jul 2011, at 16:29, Matt Glasser wrote:
> One does absolutely have to make sure that when you average your directions
> are pointing the same way.
>
> The question is: if you get the maths right and you want to average your
> images, is it conceptually correct to rotate bvecs according to
> registrations and average them or not (whether or not it makes any practical
> difference at all).
>
> Peace,
>
> Matt.
>
> -----Original Message-----
> From: FSL - FMRIB's Software Library [mailto:[log in to unmask]] On Behalf
> Of Saad Jbabdi
> Sent: Friday, July 22, 2011 8:19 AM
> To: [log in to unmask]
> Subject: Re: [FSL] averaging rotated bvecs
>
> I guess that's what Matt meant when he said "if you get the maths right"
>
> Saad.
>
>
> On 22 Jul 2011, at 14:03, Pablo Velasco wrote:
>
>> Hi Daniela,
>>
>> Intuitively, I don't think this is correct: if you average two images
> taken with diffusion gradients in exact opposite directions, the average of
> the diffusion vectors will be zero. However, the average of the images will
> not. I guess if your rotated vectors are pointing almost along the same
> direction, the error you will get from averaging directions and images will
> be very small.
>>
>> Why don't you just do the analysis on a dataset with 72 directions?
>>
>> -Pablo
>>
>
> --
> Saad Jbabdi
> University of Oxford, FMRIB Centre
>
> JR Hospital, Headington, OX3 9DU, UK
> (+44)1865-222466 (fax 717)
> www.fmrib.ox.ac.uk/~saad
>
--
Saad Jbabdi
University of Oxford, FMRIB Centre
JR Hospital, Headington, OX3 9DU, UK
(+44)1865-222466 (fax 717)
www.fmrib.ox.ac.uk/~saad
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