[Insight-developers] Can someone explain DiffusionTensor3D::GetRelativeAnisotropy() to me?

[Simon Warfield]

Torsten Rohlfing wrote:

>
> Good morning --
>
> At the risk of looking like a complete idiot, I was wondering whether 
> someone could explain the implementation of 
> DiffusionTensor3D::GetRelativeAnisotropy() to me.
>
> The code (purged of numerical safeguards etc) that I don't get is this:
>
>  const RealValueType anisotropy = 3.0 * isp - trace * trace;
>  const RealValueType relativeAnisotropySquared =
>        static_cast< RealValueType >( anisotropy / ( sqrt( 3.0 ) * 
> trace ) );
>  const RealValueType relativeAnisotropy =
>        static_cast< RealValueType >( sqrt( relativeAnisotropySquared ) );
>
> where "isp" is the inner scalar product and "trace" is the tensor 
> trace, both as returned by the respective class member functions.
>
> My problem is with the definition of "anisotropy" above. I assume that 
> this is supposed to be the inner product of the anisotropic part of 
> the tensor, that is, the inner product of  (tensor - (trace / 3) * 
> identity). This is a double sum over the squares of differences 
> between elements of the original tensor and the corresponding elements 
> of (trace/3) times the identity tensor.

Check out Equation 27 here:
http://splweb.bwh.harvard.edu:8000/pages/papers/westin/media2002/Westin_Diffusion_MedAI_2002.pdf
and the definition of the norm at the top of the column.

>
> When I write this down, expand the squared difference in the scalar 
> product and collapse the three resulting double sums, I get the inner 
> product of the tensor PLUS the square of the trace MINUS 6 times the 
> trace. This seems considerably different from what is implemented in 
> ITK, since the sign of the squared trace is inverted, and the term 
> with the trace alone is missing entirely.
>
> Could someone give me a hint what I am missing here, or maybe point me 
> to a place to read up on this?
>
> And if this was as stupid a question as I suspect, feel free to rub it 
> in ;)
>
> Best,
>  Torsten
>


-- 
Simon K. Warfield, Ph.D. warfield at bwh.harvard.edu Phone:617-732-7090
http://www.spl.harvard.edu/~warfield           FAX:  617-582-6033
Associate Professor of Radiology,          Harvard Medical School
Director, Computational Radiology Laboratory
Thorn 329, Dept Radiology,  Brigham and Women's Hospital 
75 Francis St, Boston, MA, 02115
MA 280, Dept Radiology, Children's Hospital Phone: 617-355-4566