[Insight-developers] NIfTI, DiffusionTensor3D Pixels, SymmetricSecondRankTensor

[Tom Vercauteren]

Hi Kent,

Using two loops should make it right. It would avoid constructing an
ordering vector.

for i=0:N
  for j=i:N
    NIfTIPixel[j*(j+1)/2+i] = ITKPixel[j+i*(2N-i-1)/2];
  end
end

You might want to really check these indexes though. Anyhow it should
be along these lines.

IIRC for a lower triangular matrix M stored in C syle order and i>=j you have
  M(i,j) = M[ j + i(i+1)/2 ]

Similarly for an upper triangular matrix M stored in C syle order and
j>=i you have
  M(i,j) = M[ j + i(2N-i-1)/2 ]

Here since you have symmetric matrices you also use the fact that
M(i,j) = M(j,i) to get things right when j>=i instead of i>=j or i>=j
instead of j>=i.

Hope this helps,
Tom

On Wed, Mar 11, 2009 at 21:08, kent williams
<norman-k-williams at uiowa.edu> wrote:
> I've managed to get myself confused and throw myself on the mercy of the
> more mathematically inclined.
>
> Luke logged this bug http://public.kitware.com/Bug/view.php?id=7485 a while
> back.  In essence the problem is that ITK and NIfTI disagree about the order
> of values when storing a DiffusionTensor3D, and one presumes
> SymmetricSecondRankTensor as well. ITK stores upper-triangular and NIfTI
> stores lower-triangular.
>
> In the case of DiffusionTensor3D, a symmetric 3x3 matrix, the ordering for a
> matrix
>
> A B C
> B D E is A B C D E F for ITK, and A B D C E F for NIfTI.
> C E F
>
> Given how the NIfTI filter works when it's copying pixels to and from ITK
> images, I handle this with an ordering array.  In other words I have an
> array
>    int niftiorder[9] = { 0,1,3,2,4,5,6 };
> and in essence, NIfTIPixel[i] = ITKPixel[order[i]];
>
> That's good for DiffusionTensor3D, but the general case of
> SymmetricSecondRankTensor is baffling me. For a 4x4 symmetric matrix
>  int niftiorder[10] = { 0, 1, 4, 2, 5, 7, 3, 6, 8, 9 };
>
> and 5x5 is even crazier.
>  int niftiorder[15] = { 0, 1, 5, 2, 6, 9, 3, 7, 10, 2, 4, 8, 11, 13, 14 };
>
> So I'm not seeing how I would go about deriving this ordering.  I.e. I don't
> know what general purpose code would handle re-ordering the scalars from an
> upper triangular to a lower triangular or vice verse.
>
> Does anyone have a suggestion on how to go about this?
>
> And while I'm on the subject, why is the VNL Symmetric Matrix lower
> triangular, yet ITK is upper triangular?
>
>
>
>
>
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