# 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 $$\mathbf{b}$$: $$b = \mathrm{Tr}(\mathbf{b}) = 3b_\mathrm{iso}$$.