Fig. 1.

This figure serves the double purpose of explaining the intra-volume movement problem as well as our forward spatial model. To understand the problem, consider a sequential single-slice sequence where slices 15 and 16 have just been acquired, then the subjects move (“looking up into the sky”) so that slices 17 and 18 are acquired at the positions indicated in the schematic on the left. The reconstruction process does not know that the subject has moved and will stack the slices on top of each other as seen on the right hand side. If we assume that the subject stays in this new position for the remaining slices the apparent shape of the brain will now be as seen on the right (looking more like a sperm whale than a brain), as opposed to the true shape shown on the left. In order to understand the forward model, assume that all movement is known such that we can accurately calculate the matrices $\mathbf{R}(\mathbf{r}(15))$ and $\mathbf{R}(\mathbf{r}(18))$. The image ${\widehat{f}}_i$ from Eq. (1) serves as the image on the left, and is hence “known”. The aim of the forward model is now to calculate the image on the right given that we know $\mathbf{B}$, i.e. all the movements. This is performed using the following strategy: for all coordinates $\mathbf{x}$ on the right calculate $\mathbf{x}\prime$, map $\mathbf{x}\prime$ into the regular grid of ${\widehat{f}}_i$ and use standard spline interpolation to calculate an intensity ${\widehat{f}}_i(\mathbf{x}\prime )$ that is written into $f_i(\mathbf{x})$ on the right. Using this strategy it is possible to predict the “observed” image for any set of movements $\mathbf{B}$.