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):
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]:
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
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:
where the bounds of \(\lambda_\eta\) ensure the positivity of \(\lambda_{XX}\) and \(\lambda_{YY}\) when \(\lambda_\Delta > 0\). Equivalently, one has
Transfer between the previous representations of \(\mathbf{\Lambda}\) is ensured via Euler rotation according to
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
with
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
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
with
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 \(\mathbf{b}\): \(b = \mathrm{Tr}(\mathbf{b}) = 3b_\mathrm{iso}\).
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