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

[Torsten Rohlfing]

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.

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

-- 
Torsten Rohlfing, PhD          SRI International, Neuroscience Program
 Research Scientist             333 Ravenswood Ave, Menlo Park, CA 94025
  Phone: ++1 (650) 859-3379      Fax: ++1 (650) 859-2743
   torsten at synapse.sri.com        http://www.stanford.edu/~rohlfing/

     "Though this be madness, yet there is a method in't"