Tensor Classification

Geometrically, a diffusion tensor can be thought of as an ellipsoid with its three axes oriented along the tensor's three perpendicular eigenvectors, with the three semi-axis lengths proportional to the square root of eigenvalues of the tensor - mean diffusion distances [1]. In general, eigenvalues $\lambda$ and eigenvectors $\mathbf e$ can be found as a solution to the eigen-equation

$${\mathbf T} {\mathbf e}_i = \lambda_i {\mathbf e}_i$$ (2)

Since the tensor is symmetric, its eigenvalues are always real numbers, and the eigenvectors are orthogonal and form a Cartesian vector basis $\{{\mathbf e}_1, {\mathbf e}_2, {\mathbf e}_3\}$. This basis (frame of reference) can be used to represent the tensor in diagonal form and to specify directions with respect to the ``world coordinate'' system

$${\mathbf T} = \{{\mathbf e}_1,{\mathbf e}_2,{\mathbf e}_3\} \left... ...lambda_3 \end{array} \right) \{{\mathbf e}_1, {\mathbf e}_2, {\mathbf e}_3 \}^T$$ (3)

Using the ellipsoidal interpretation, one can classify the diffusion properties of tissue according to the shape of the ellipsoids, with extended ellipsoids corresponding to regions with strong linear diffusion (long, thin cells) flat ellipsoids to planar diffusion, and spherical ellipsoids to regions of isotropic media (such as fluid-filled regions like the ventricles). The quantitative classification can be done through the coefficients $c_\ell,c_p,c_s$ (linear, planar, spherical) proposed in [23,22]:

$$c_\ell = \frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2 +\lambda_3}$$ (4)

$$c_p = \frac{2(\lambda_2 - \lambda_3)}{\lambda_1 + \lambda_2 +\lambda_3}$$ (5)

$$c_s = \frac{3\lambda_3}{\lambda_1 + \lambda_2 +\lambda_3}$$ (6)

These coefficients are normalized to the range of $[0..1]$ and could be interpreted as barycentric coordinates. For example, close to $1$ values of $c_\ell$ chooses the regions with strong linear $(\lambda_1 >> \lambda_2 \approx \lambda_3)$ diffusion.