t_0)^2(\overline{m}-m_i)] \overset{!}{=} 0.\end{aligned}$$ Thus, $$\overline{m} = \frac{\sum_i (t_i - t_0)^2m_i}{\sigma^2 + \sum_i (t_i - t_0)^2}.\label{eq:gr_m0}$$ In practice, $\sigma^2$ is very small and can thus be omitted. Furthermore, $m_i$ is obtained by either registering $\overline{I}$ to $Y^i$ in unit time or, as in our FPSGR approach, by predicting the momenta $m_i$ via FPIR, denoted as $\widetilde{m}_i$. As Equation \[eq:gr\_m0\] was derived assuming that images are transformed into each other in time $t_i-t_0$ instead of unit time, the obtained unit-time predicted momenta $\widetilde{m}_i$ correspond in fact to the approximation $\widetilde{m}_i\approx (t_i-t_0)m_i$. Finally, we obtain the approximated optimal $\overline{m}$ of the energy functional in Eq. , for a fixed $\overline{I}=I_0$ as $$\begin{aligned}
\qquad \qquad \overline{m} \approx \frac{\sum_i (t_i-t_0)\widetilde{m}_i}{\sum_i (t_i-t_0)^2}\label{eq:sgr}.\end{aligned}$$

Setup / Training {#sec:experimental_setup}
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All our experiments use 3D images from the `ADNI` dataset[^1] which consists of 6471 3D MR brain images of size $220 \times 220 \times 220$ voxels. In particular, `ADNI-1` contains 3479 images from 833 subjects and `ADNI-2` contains 2992 images from 823 subjects. Images belong to various types of diagnostic categories which we will discuss later. 1ex We perform the following two types of studies:

Registration

:   We assess our hypothesis that training FPIR on longitudinal data for longitudinal 