Tensor parameterization

General case

Let us introduce an arbitrary symmetric semipositive-definite tensor \(\mathbf{\Lambda}\) in a general frame of reference (e.g. that of the lab):

\[ \mathbf{\Lambda} = \begin{pmatrix} \lambda_{xx} & \lambda_{xy} & \lambda_{xz} \\ \lambda_{yx} & \lambda_{yy} & \lambda_{yz} \\ \lambda_{zx} & \lambda_{zy} & \lambda_{zz} \end{pmatrix} = \begin{pmatrix} \lambda_{xx} & \lambda_{xy} & \lambda_{xz} \\ \cdot & \lambda_{yy} & \lambda_{yz} \\ \cdot & \cdot & \lambda_{zz} \end{pmatrix} \, , \]

where symmetry imposes that \(\lambda_{ij} = \lambda_{ji}\), yielding six independent elements in total. Upon diagonalization, \(\mathbf{\Lambda}\) can be expressed in its eigenbasis (basis of eigenvectors) using the Haeberlen convention [@Haeberlen:1976]:

\[ \mathbf{b}^{(\mathrm{diag})} = \begin{pmatrix} \lambda_{XX} & 0 & 0 \\ 0 & \lambda_{YY} & 0 \\ 0 & 0 & \lambda_{ZZ} \end{pmatrix} \quad \mathrm{with} \quad \left\vert \lambda_{YY} - \lambda_\mathrm{iso} \right\vert \leq \left\vert \lambda_{XX} - \lambda_\mathrm{iso} \right\vert \leq \left\vert \lambda_{ZZ} - \lambda_\mathrm{iso} \right\vert , \]

where \(\lambda_\mathrm{iso} = \mathrm{Tr}(\mathbf{\Lambda})/3\) is the average of \(\mathbf{\Lambda}\)'s eigenvalues (also called “isotropic average”). This ordering convention assures that \(\lambda_{ZZ}\) is furthest from the isotropic average while \(\lambda_{YY}\) is closest. In particular, the eigenvector associated with \(\lambda_{ZZ}\) is, therefore, the main eigenvector of \(\mathbf{\lambda}\).

\(\mathbf{\Lambda}\) can also be written in an eigenbasis form that directly reflects its size and shape, namely

\[ \mathbf{\Lambda}^{(\mathrm{diag})} = \lambda_\mathrm{iso} \left\{ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} + \lambda_\Delta \left[ \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 2 \end{pmatrix} +\lambda_\eta \begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix} \right] \right\} , \]

where \(\lambda_\Delta\) and \(\lambda_\eta\) denote \(\mathbf{\Lambda}\)'s normalized anisotropy and asymmetry, respectively. These shape parameters are related to the eigenvalues \(\lambda_{XX}\), \(\lambda_{YY}\) and \(\lambda_{ZZ}\) as follows:

\[ \begin{aligned} \lambda_\Delta & = \frac{1}{b}\left( \lambda_{ZZ} - \frac{\lambda_{XX} + \lambda_{YY}}{2} \right) \in [-0.5,1]\, , \\ \lambda_\eta & = \frac{1}{2}\, \frac{\lambda_{YY} - \lambda_{XX}}{\lambda_\mathrm{iso}\lambda_\Delta}\in\left[ \mathrm{max}\!\left(-\left\vert 1-\frac{1}{\lambda_\Delta} \right\vert,-1 \right)\, ,\, \mathrm{min}\!\left(\left\vert 1-\frac{1}{\lambda_\Delta} \right\vert,1 \right) \right] \, , \end{aligned} \]

where the bounds of \(\lambda_\eta\) ensure the positivity of \(\lambda_{XX}\) and \(\lambda_{YY}\) when \(\lambda_\Delta > 0\). Equivalently, one has

\[ \begin{aligned} \lambda_{XX} & = \lambda_\mathrm{iso} [1-\lambda_\Delta(1+\lambda_\eta)]\, , \\ \lambda_{YY} & = \lambda_\mathrm{iso} [1-\lambda_\Delta(1-\lambda_\eta)]\, , \\ \lambda_{ZZ} & = \lambda_\mathrm{iso} [1+2\lambda_\Delta] \, . \end{aligned} \]

Transfer between the previous representations of \(\mathbf{\Lambda}\) is ensured via Euler rotation according to

\[ \mathbf{\Lambda}= \mathbf{R}_\mathrm{Euler}(\alpha,\beta,\gamma) \cdot \mathbf{\Lambda}^{(\mathrm{diag})} \cdot \mathbf{R}_\mathrm{Euler}^{\mathrm{T}}(\alpha,\beta,\gamma) \, , \]

where the superscript “\(\mathrm{T}\)” indicates transposition and the Euler rotation matrix \(\mathbf{R}_\mathrm{Euler}(\alpha,\beta,\gamma)\) depends on the three Euler angles according to

\[ \mathbf{R}_\mathrm{Euler}(\alpha,\beta,\gamma) = \mathbf{R}_z(\gamma)\cdot\mathbf{R}_y(\beta)\cdot\mathbf{R}_z(\alpha) \, , \]

with

\[ \begin{aligned} \mathbf{R}_y(\theta) & = \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{pmatrix} \, , \\ \mathbf{R}_z(\theta) & = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \, , \end{aligned} \]

for any arbitrary angle \(\theta\).

While the normalized anisotropy \(\lambda_\Delta\) was first introduced in the context of diffusion tensors in Ref.[@Conturo:1996], subsequently related to other measures of diffusion anisotropy [@Kingsley:2006], it was first applied to b-tensors in Ref.[@Eriksson:2015].

Axisymmetric case

The axisymmetric case corresponds to \(\lambda_\eta = 0\). In this context, the number of independent elements in \(\mathbf{\Lambda}\) reduces to four. Indeed, \(\lambda_\eta\) is set to zero and only two Euler angles are now necessary to describe \(\mathbf{\Lambda}\)'s main orientation. Using previous equations, an axisymmetric tensor is parametrized as

\[ \mathbf{\Lambda}^{(\mathrm{diag})} = \begin{pmatrix} \lambda_\mathrm{iso}(1-\lambda_\Delta) & 0 & 0 \\ 0 & \lambda_\mathrm{iso}(1-\lambda_\Delta) & 0 \\ 0 & 0 & \lambda_\mathrm{iso}(1+2\lambda_\Delta) \end{pmatrix} = \begin{pmatrix} \lambda_\perp & 0 & 0 \\ 0 & \lambda_\perp & 0 \\ 0 & 0 & \lambda_\parallel \end{pmatrix} \, , \]

where \(\lambda_\parallel\) and \(\lambda_\perp\) are the axial and radial eigenvalues, respectively. Moreover, transfer between the previous representations of \(\mathbf{\Lambda}\) in the axisymmetric case is ensured via

\[ \mathbf{\Lambda} = \mathbf{R}(\theta,\phi) \cdot \mathbf{\Lambda}^{(\mathrm{diag})} \cdot \mathbf{R}^{\mathrm{T}}(\theta,\phi)\, , \]

with

\[ \mathbf{R}(\theta,\phi) = \mathbf{R}_z(\phi)\cdot\mathbf{R}_y(\theta) \, , \]

using the rotation matrices given above.

While the isotropic diffusivity corresponds to the isotropic average of the diffusion tensor, \(D_\mathrm{iso} = \mathrm{Tr}(\mathbf{D})/3\), the b-value is directly given by the trace of the b-tensor \(\mathbf{b}\): \(b = \mathrm{Tr}(\mathbf{b}) = 3b_\mathrm{iso}\).