# Signal representation

## Generalized two-term cumulant expansion

Published in Ref.[@Westin:2016], q-space trajectory imaging (QTI) fits the multidimensional diffusion signal using the following generalized two-term cumulant expansion:

$\mathcal{S}(\mathbf{b}) = \mathcal{S}_0\,\exp\!\left(-\mathbf{b}:\langle \mathbf{D}\rangle + \frac{1}{2}\, \mathbf{b}^{\otimes 2}:\mathbb{C}\right) ,$

where $$\mathbf{b}$$ is the second-order b-tensor, $$\langle \mathbf{D} \rangle$$ is the second-order mean diffusion tensor,

$\mathbb{C} = \langle \mathbf{D}^{\otimes 2} \rangle - \langle \mathbf{D} \rangle^{\otimes 2}$

is the fourth-order covariance tensor, and $$\mathbf{D}^{\otimes 2} = \mathbf{D}\otimes\mathbf{D}$$ denotes the outer tensor product of $$\mathbf{D}$$ with itself. Sometimes called “covariance tensor approximation”, QTI can equivalently be thought of as assuming that the intra-voxel distribution of diffusion tensors $$\mathcal{P}(\mathbf{D})$$ is well captured by a normal distribution of tensors.

## Voigt notation

The symmetry of all previous tensors allows for drastic simplifications during implementation using the Voigt notation

$\mathbf{D} = \begin{pmatrix} D_{xx} & D_{xy} & D_{xz} \\ \cdot & D_{yy} & D_{yz} \\ \cdot & \cdot & D_{zz} \end{pmatrix} \equiv \mathbf{d}_\mathrm{Voigt} = \begin{pmatrix} D_{xx} & D_{yy} & D_{zz} & \sqrt{2}\, D_{yz} & \sqrt{2}\, D_{xz} & \sqrt{2}\, D_{xy} \end{pmatrix}^\mathrm{T}\, .$

Indeed, this notation allows to write the $$\ 3\times 3 \times 3 \times 3$$ tensor $$\mathbf{D}^{\otimes 2}$$ as the following $$\ 6\times 6$$ tensor: $$\mathbf{D}^{\otimes 2} \equiv \mathbf{d}_\mathrm{Voigt}\cdot\mathbf{d}_\mathrm{Voigt}^\mathrm{T}$$. Moreover, these symmetries imply that the mean diffusion tensor possesses 6 independent elements, and that the covariance tensor possesses 21 independent elements:

$\mathbb{C} = \begin{pmatrix} C_{xx,xx} & C_{xx,yy} & C_{xx,zz} & \sqrt{2}\, C_{xx,yz} & \sqrt{2}\, C_{xx,xz} & \sqrt{2}\, C_{xx,xy} \\ \cdot & C_{yy,yy} & C_{yy,zz} & \sqrt{2}\, C_{yy,yz} & \sqrt{2}\, C_{yy,xz} & \sqrt{2}\, C_{yy,xy} \\ \cdot & \cdot & C_{zz,zz} & \sqrt{2}\, C_{zz,yz} & \sqrt{2}\, C_{zz,xz} & \sqrt{2}\, C_{zz,xy} \\ \cdot & \cdot & \cdot & 2\, C_{yz,yz} & 2\, C_{yz,xz} & 2\, C_{yz,xy} \\ \cdot & \cdot & \cdot & \cdot & 2\, C_{xz,xz} & 2\, C_{xz,xy} \\ \cdot & \cdot & \cdot & \cdot & \cdot & 2\, C_{xy,xy} \\ \end{pmatrix}\, ,$

where $$C_{ij,kl} = \langle D_iD_j\rangle - \langle D_k\rangle \langle D_l \rangle$$ with $$i,j,k,l\in \{x,y,z\}$$. Therefore, with the addition of $$\mathcal{S}_0$$, QTI consists in a 28-dimensional fitting procedure.

## A lightening fast inversion

While a 28-dimensional fit seems daunting, it can be made incredibly fast upon taking the logarithm of the QTI signal and writing it as

$\underbrace{ \begin{pmatrix} \log \mathcal{S}_1 \\ \vdots \\ \log \mathcal{S}_{N_\mathrm{acq}} \end{pmatrix} }_{\mathbf{S}\in\mathbb{R}^{N_\mathrm{acq}\times 1}} = \underbrace{ \begin{pmatrix} 1 & -\mathbf{b}_{\mathrm{Voigt},1}^\mathrm{T} & \frac{1}{2}\, \tilde{b}_{\mathrm{Voigt},1}^\mathrm{T} \\ \vdots & \vdots & \vdots \\ 1 & -\mathbf{b}_{\mathrm{Voigt},N_\mathrm{acq}}^\mathrm{T} & \frac{1}{2}\, \tilde{b}_{\mathrm{Voigt},N_\mathrm{acq}}^\mathrm{T} \end{pmatrix} }_{\mathbf{X}\in\mathbb{R}^{N_\mathrm{acq}\times 28}} \cdot \underbrace{ \begin{pmatrix} \mathcal{S}_0 \\ \langle \mathbf{d} \rangle_\mathrm{Voigt} \\ \tilde{c}_\mathrm{Voigt} \end{pmatrix} }_{\mathbf{\beta}\in\mathbb{R}^{28\times 1}} \, ,$

where $$N_\mathrm{acq}$$ is the number of acquisition points, $$\mathbf{b}_\mathrm{Voigt}$$ and $$\langle\mathbf{d} \rangle_\mathrm{Voigt}$$ are the $$\ 6\times 1$$ versions of $$\mathbf{b}$$ and $$\langle\mathbf{D}\rangle$$, respectively, and $$\tilde{b}_\mathrm{Voigt}$$ and $$\tilde{c}_\mathrm{Voigt}$$ are the $$\ 21\times 1$$ versions of $$\mathbf{b}^{\otimes 2}$$ and $$\mathbb{C}$$, respectively. The $$\mathbf{\beta}$$ vector can now be estimated by mere pseudoinversion according to

$\mathbf{\beta} = (\mathbf{X}^\mathrm{T}\cdot \mathbf{X})^{-1} \cdot\mathbf{X}^\mathrm{T}\cdot\mathbf{S} \, .$

## Projection tensors

Introducing $$\mathbf{E}_\mathrm{iso} = \mathbf{I}_3/3$$ and $$\mathbb{E}_\mathrm{iso} = \mathbf{I}_6/3$$, where $$\mathbf{I}_n$$ is the $$n\times n$$ identity matrix, the definition of QTI's heterogeneity metrics relies on inner products of the covariance tensor with the following projection tensors:

$\mathbb{E}_\mathrm{bulk} = \mathbf{E}_\mathrm{iso}^{\otimes 2} \qquad , \qquad \mathbb{E}_\mathrm{shear} = \mathbb{E}_\mathrm{iso} - \mathbb{E}_\mathrm{bulk} \, ,$

by analogy with the bulk and shear modulus of the fourth-order stress tensor in mechanics. In Voigt notation, these projection tensors write

\begin{aligned} \mathbb{E}_\mathrm{bulk} & = \frac{1}{9} \begin{pmatrix} 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} \, , \\ \mathbb{E}_\mathrm{shear} & = \frac{1}{9} \begin{pmatrix} 2 & -1 & -1 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 \\ -1 & -1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 3 \end{pmatrix} \, . \end{aligned}

They satisfy the following relationships:

\begin{aligned} \langle \mathbf{D}\rangle^{\otimes 2} : \mathbb{E}_\mathrm{bulk} & = E_\lambda[\langle \mathbf{D} \rangle]^2 \, , \\ \langle \mathbf{D}^{\otimes 2}\rangle : \mathbb{E}_\mathrm{bulk} & = \mathrm{E}[E_\lambda[\mathbf{D}]^2] \, , \\ \langle \mathbf{D}\rangle^{\otimes 2} : \mathbb{E}_\mathrm{shear} & = V_\lambda (\langle \mathbf{D}\rangle) \, , \\ \langle \mathbf{D}^{\otimes 2}\rangle : \mathbb{E}_\mathrm{shear} & = \mathrm{E}[ V_\lambda (\mathbf{D})] \, , \end{aligned}

where $$E_\lambda (\mathbf{T})$$ and $$V_\lambda (\mathbf{T})$$ are the eigenvalue expectation and variance of any arbitrary tensor $$\mathbf{T}$$, respectively. Note that we use $$\langle \,\cdot\, \rangle$$ and $$\mathrm{E}[ \,\cdot\, ]$$ for matrix-valued and scalar averages over the voxel content, respectively, for clarity. Besides, one has

$\langle \mathbf{D}^{\otimes 2}\rangle = \mathbb{C} + \langle \mathbf{D}\rangle^{\otimes 2} \, ,$

by definition of the covariance tensor.

## Simplification as diffusion tensor imaging (DTI)

Diffusion tensor imaging (DTI) [@Basser:1994] corresponds to QTI, albeit with a zero covariance tensor $$\mathbb{C} = 0$$. Naturally, this simplification implies that no heterogeneity measure can be extracted from DTI.