Fig. 3.

Again, this figure uses the same acquisition situation as in Fig. 1. We wish to use the slices in ${\mathcal{S}}_o$ to re-create a volume in ${\mathcal{S}}_m$. In order to do so we need to know what $x\prime$ and $y\prime$ coordinates in the original slice s in ${\mathcal{S}}_o$ correspond to integer/regular coordinates x and y in ${\mathcal{S}}_m$. We also need to know what non-integer coordinate $z^{*}$ in ${\mathcal{S}}_m$ corresponds to every combination of x, y and s. That leads to the equation in the figure, where there are unknowns on both sides ($x\prime$, $y\prime$ in ${\mathcal{S}}_o$ and $z^{*}$ in ${\mathcal{S}}_m$).