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Tensor Invariants

Tensor invariants (rotational invariants) are combinations of tensor elements that do not change after the rotation of the tensor's frame of reference, and thus do not depend on the orientation of the patient with respect to the scanner when performing DT imaging. The well known invariants are the eigenvalues of diffusion tensor (matrix) $ D$, which are the roots of corresponding characteristic equation

$\displaystyle \lambda^3 - C_1 \cdot \lambda^2 + C_2 \cdot \lambda - C_3 = 0,$ (3)

with coefficients
$\displaystyle C_1$ $\displaystyle =$ $\displaystyle D_{xx} + D_{yy} + D_{zz}$  
$\displaystyle C_2$ $\displaystyle =$ $\displaystyle D_{xx} D_{yy} -D_{xy}D_{yx} + D_{xx}D_{zz} - D_{xz}D_{zx} + D_{yy}D_{zz} -
D_{yz}D_{zy}$ (4)
$\displaystyle C_3$ $\displaystyle =$ $\displaystyle D_{xx}(D_{yy} D_{zz} - D_{zy} D_{yz})$  
  $\displaystyle -$ $\displaystyle D_{xy}(D_{yx} D_{zz} - D_{zx} D_{yz}) + D_{xz}(D_{yx} D_{zy} -D_{zx}
D_{yy}).$  

Since the roots of Equation (3) are rotational invariants, the coefficients $ C_1$, $ C_2$ and $ C_3$ are also invariant. In the eigen- frame of reference they can be easily expressed through the eigenvalues
$\displaystyle C_1$ $\displaystyle =$ $\displaystyle \lambda_1 + \lambda_2 + \lambda_3$  
$\displaystyle C_2$ $\displaystyle =$ $\displaystyle \lambda_1\lambda_2 + \lambda_1\lambda_3 + \lambda_2\lambda_3$ (5)
$\displaystyle C_3$ $\displaystyle =$ $\displaystyle \lambda_1 \lambda_2 \lambda_3.$  

and are proportional to the sum of the radii, surface area and the volume of the ``diffusion'' ellipsoid. Then instead of using $ (\lambda_1,\lambda_2,\lambda_3)$ to describe the dataset, we can use $ (C_1,C_2,C_3)$. Moreover, since $ C_i$ are the coefficients of characteristic equation, they are less sensitive to noise, then roots $ \lambda_i$ of the same equation.

Any combination of the above invariants is, in turn, an invariant. We consider the following dimensionless combination: $ C_1 C_2/C_3$. In the eigenvector frame of reference it becomes

$\displaystyle \frac{C_1 C_2}{C_3} = 3 + \frac{\lambda_2 + \lambda_3}{\lambda_1}...
...rac{\lambda_1 + \lambda_3}{\lambda_2} + \frac{\lambda_1 + \lambda_2}{\lambda_3}$ (6)

and we can define a new dimensionless anisotropy measure

$\displaystyle C_a = \frac{1}{6}\left[\frac{C_1 C_2}{C_3} - 3 \right].$ (7)

It is easy to show that for isotropic diffusion, when $ \lambda_1 =
\lambda_2 = \lambda_3$, the coefficient $ C_a = 1$. In the anisotropic case, this measure is identical for both linear, directional diffusion $ (\lambda_1 »
\lambda_2 \approx \lambda_3)$ and planar diffusion $ (\lambda_1 \approx
\lambda_2 » \lambda_3)$ and is equal to

$\displaystyle C_a^{limit} \approx \frac{1}{3}\left[1 + \frac{\lambda_1}{\lambda_3} + \frac{\lambda_3}{\lambda_1}\right].$ (8)

Thus $ C_a$ is always $ \sim \lambda_{max}/\lambda_{min}$ and measures the magnitude of the diffusion anisotropy. We again want to emphasize, that we use eigenvalue representation here only to analyze the behavior of the coefficient $ C_a$, but we use invariants $ (C_1,C_2,C_3)$ to compute it using Equations (5) and (7).

Figure 2:
\includegraphics[width=7in]{fig2.eps}

Figure 2. Isotropic $ C_1$ (left) and anisotropic $ C_a$ (right) tensor invariants for the tensor slice shown in Figure 1.

 



Next: Geometric Modeling Up: Level Set Modeling and Previous: Introduction
Leonid Zhukov 2003-09-14