Diffusion Indices

Suppose \underline{D} is a second order diffusion tensor (3x3 matrix) with eigenvalues \lambda\_i at a given voxel then several kinds of anisotropy measures are defined as follows:

Linear Anisotropy

LA :=_{df} \frac{\lambda_1 -\lambda_2}{\sqrt{\lambda_1^2+\lambda_2^2+\lambda_3^2} }

Fractional Anisotropy

See (http://dx.doi.org/10.1006/jmrb.1996.0086) for definition:

FA :=_{df} \sqrt{\frac{1}{2} } \frac{\sqrt{(\lambda_1-\lambda_2)^2 + (\lambda_2-\lambda_3)^2 + (\lambda_3-\lambda_1)^2} }{\sqrt{\lambda_1^2+\lambda_2^2+\lambda_3^2} }

This is identical to (http://rsl.stanford.edu/moseley/tensorcalc/tensorcalc/Output/FA.html):

\frac{\sqrt{3} }{\sqrt{2} }\frac{\sqrt{(\lambda_1-\lambda)^2+(\lambda_2-\lambda)^2+(\lambda_3-\lambda)^2} }{\sqrt{\lambda_1^2+\lambda_2^2+\lambda_3^2} }

with \lambda = \frac{1}{3} tr(\underline{D}) = \frac{1}{3}(\lambda_1+\lambda_2+\lambda_3)