the following.

LDDMM
-----

Shooting-based LDDMM and geodesic regression minimize $$\begin{aligned}
\label{eq:gr}
E(I_0, m_0) = \frac{1}{2}\langle m_0, Km_0\rangle + \frac{1}{\sigma^2}\sum_i d^2(I(t_i), Y^i), \\ 
\notag
s.t. \quad m_t + \text{ad}^*_vm = 0, I_t + \nabla I^Tv = 0, m - Lv = 0,\end{aligned}$$ where $I_0$ is the initial image (known for image-to-image registration and to be determined for geodesic regression), $m_0$ is the initial momentum, $K$ is a smoothing operator that connects velocity $v$ and momentum $m$ as $v = Km$ and $m = Lv$ with $K = L^{-1}$, $\sigma > 0$ is a weight, $Y^i$ is the measured image at time $t_i$ (there will be only one such image for image-to-image registration at $t=1$), and $d^2(I_1, I_2)$ denotes the image similarity measure between $I_1$ and $I_2$ (for example $L_2$ or geodesic distance); $\text{ad}^*$ is the dual of the negative Jacobi-Lie bracket of vector fields: $\text{ad}_vw = -[v, w] = Dvw - Dwv$ and $D$ denotes the Jacobian. The deformation of the source image $I_0\circ\Phi^{-1}$ can be computed by solving $\Phi^{-1}_t + D\Phi^{-1}v=0,~\Phi^{-1}(0)=\text{id}$, where $\text{id}$ denotes the identity map.

FPIR
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Fast predictive image registration [@ref:yang2016; @quicksilver] aims at predicting the initial momentum, $m_0$, between a source and a target image 