[Mrtrix-discussion] Probabilistic part of SD_PROB

[Donald Tournier]

Hi René,

That's an interesting question, and there's quite a few ways look at
it. First off, you're right that by using a bootstrapping procedure on
the peak orientation does provide a probability density function.
However, you have to bear in mind that this PDF is based on the
implicit assumption that fibre orientations are discrete. In other
words, that each fibre population runs in a single well-defined
orientation. This obviously does not allow for any spread in the fibre
orientations, either due to curvature or 'fanning'. On the other hand,
if you can make that assumption, then the bootstrap procedure does
indeed provide a much tighter PDF than that provided by CSD, and is
therefore a very attractive method. If you're interested, Ben
Jeurissen has just had a paper accepted in NeuroImage on that topic,
in which we discuss that particular issue.

On the other hand, if you're interested in a more general
representation of the FOD, you'll need to relax the assumption of
discrete orientations. In this case, the representation of the FOD
information goes from a set of discrete orientations to a continuous
function, best represented using some appropriate set of basis
functions, such as spherical harmonics. Whichever basis set is used,
this inherently introduces a limit on the achievable resolution (as
you pointed out). The limit on the angular resolution is probably more
a function of the data than the particular algorithm used (I had an
ISMRM abstract about that last year). I believe CSD provides the best
estimate of the FOD when the FOD is assumed continuous.

You also mentioned that in your opinion the FOD itself does not
contain any true probability information. I have to disagree with
that. Even if the FOD could be estimated perfectly, such that its
parameters are know with absolute precision, it still provides a
suitable PDF for use in probabilistic tracking, if viewed from the
following perspective. The FOD provides an estimate of the 'amount' of
fibres as a function of their orientation, within the volume of
interest (i.e. the voxel). It does not provide any information as to
where in the voxel fibres with a particular orientation might be.
Therefore, if you select a point at random from within the voxel of
interest, the probability of the fibres at that location being
oriented along a particular direction is directly proportional to the
FOD amplitude along that direction. This is precisely what is required
for probabilistic tracking. Obviously, in practice the FOD can't be
estimated with anything approaching absolute precision - there will
always be some noise and relatively poor angular resolution. But the
argument holds nonetheless.

Something else that you might have had in mind is to somehow generate
a 'probabilistic' CSD algorithm, using the bootstrap for example to
generate a PDF over the parameters of the FOD. While this is possible,
given the argument above it seems to me that this amounts to
calculating the PDF of a PDF, which to my mind is overkill. If you
have a reasonable estimate of the PDF, that should be sufficient to
sample from with good accuracy.

Anyway, I hope that's answered your question...
Cheers,

Donald.


2010/2/17 René Besseling <r.m.h.besseling at gmail.com>:
> Dear Donald,
>
>
>
> I was wondering what the real probabilistic part of SD_PROB is. The FOD
> glyphs themselves do not contain any true probabilistic information: the
> lobes of these glyphs have certain widths because they are only defined up
> until a certain resolution (a certain number of spherical harmonics). The
> width of a lobe is therefore not a pure measure for the uncertainty that a
> fiber population is oriented in that direction. To draw a parallel with the
> diffusion tensor model: if we assume the main eigenvector e_1 of the DT
> gives the direction of the local fiber population, by bootstrapping we can
> collect several e_1 estimates. If the direction of each e_{1,i} is given by
> (\phi_i,\theta_i), we can uses these e_1 estimates to estimate
> P(\phi_i,\theta_i). When doing fiber tracking, in every propagation step we
> choose a certain local propagation direction (\phi_i,\theta_i) and we can
> calculate the product of P(\phi_i,\theta_i) for all voxels i along the track
> to get a measure of the track probability. How do this translate to fiber
> tracking based on CSD based FOD glyphs?
>
>
>
> Best regards,
>
>
>
> René Besseling
>
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-- 
Jacques-Donald Tournier (PhD)
Brain Research Institute, Melbourne, Australia
Tel: +61 (0)3 9496 4078