Generalised Q-Sampling Imaging

These notes are to help the user of the DIPY module understand Frank Yeh’s Generalised Q-Sampling Imaging (GQI) [reference?].

The starting point is the classical formulation of joint k-space and q-space imaging (Calaghan 8.3.1 p. 438) using the narrow pulse gradient spin echo (PGSE) sequence of Tanner and Stejskal:

\[S(\mathbf{k},\mathbf{q}) = \int \rho(\mathbf{r}) \exp [j 2 \pi \mathbf{k} \cdot \mathbf{r}] \int P_{\Delta} (\mathbf{r}|\mathbf{r}',\Delta) \exp [j 2 \pi \mathbf{q} \cdot (\mathbf{r}-\mathbf{r'})] \operatorname{d}\mathbf{r}' \operatorname{d}\mathbf{r}.\]

Here \(S\) is the (complex) RF signal measured at spatial wave number \(\mathbf{k}\) and magnetic gradient wave number \(\mathbf{q}\).

\(\rho\) is the local spin density (number of protons per unit volume contributing to the RF signal).

\(\Delta\) is the diffusion time scale of the sequence.

\(P_{\Delta}\) is the averages diffusion propagator (transition probability distribution).