Algorithm 1.

Image Matching with finite-dimensional parameterization

1:	αi(t) ← 0 for all i = 1, …, N and all t
2:	c0 ← initial positions of control points (input)
3:	repeat {Gradient descent}
4:	{Compute path of control points (forward integration)}
5:	$c_i(t)=c_i(0)+{\int }_0^t{\sum }_{j=1}^{N}K(c_i(s),c_j(s)){\alpha }_j(s)ds$
6:	{Compute deformed source image (backward integration)}
7:	$y_k(t)=y_k(1)-{\int }_0^t{\sum }_{j=1}^{N}K(y_k(s),c_j(s)){\alpha }_j(s)ds$
8:	{Compute gradient of source image}
9:	∇yk(0)E = 2 (I0(yk(0)) - I1(yk)) ∇yk(0)I0
10:	{Compute auxiliary variable ηy(forward integration)}
11:	${\eta }_p^y(t)=-{\nabla }_{y_p(0)}E-{\int }_0^t{\sum }_{q=1}^N{\alpha }_q^t(s){\eta }_p^y(s){\nabla }_{1}K(y_p(s),c_q(s))ds$
12:	{Compute auxiliary variable ηc (backward integration)}
13:	${\eta }_i^c(t)={\int }_t^1{\sum }_{j=1}^N\left({\alpha }_j{(s)}^t{\eta }_i^c(s)+{\alpha }_i{(s)}^t{\eta }_j^c(s)+2\gamma {\alpha }_i{(s)}^t{\alpha }_j(s)\right){\nabla }_{1}K(c_i(s),c_j(s))+{\sum }_{k=1}^M{\alpha }_i{(s)}^t{\eta }_k^y(s){\nabla }_{2}K(y_k(s),c_i(s))ds$
14:	{Solve the linear system}
15:	${\sum }_{j=1}^{N}K(c_i(t),c_j(t)){\stackrel{\sim }{\eta }}_j^y(t)={\sum }_{k=1}^{M}K(c_i(t),y_k(t)){\eta }_k^y(t)$ (η̃y is of dimension 3N)
16:	{Compute gradient}
17:	${\nabla }_{i}J(t)=2\gamma {\alpha }_i(t)+{\eta }_i^c(t)+{\stackrel{\sim }{\eta }}_i^y(t)$
18:	{Update time-varying momenta}
19:	αi(t) ← αI(t) − ε∇αiJ(t)
20:	{Update initial positions of control points}
21:	$c_i(0)\leftarrow c_i(0)-\epsilon {\eta }_i^c(0)$
22:	until Convergence