ALGORITHM 1.

Block minimization scheme

while |Ek – Ek−1| < tol do

–Minimization w.r.t. $\{{\widehat{\theta }}_{in}^k,{\widehat{\theta }}_{out}^k\}$:

$\{\begin{array}{c}\text{Minimization}\mathrm{w}.\mathrm{r}.\mathrm{t}.{\theta }_{in},{\theta }_{out}\text{using}\mathrm{EM}\text{algorithm}\left[23\right]:\hfill \\ \{{\widehat{\theta }}_{in}^{k+1},{\widehat{\theta }}_{out}^{k+1}\}\leftarrow \mathrm{EM}\left(\{{\widehat{\varphi }}^k,{\widehat{\mathbf{w}}}^k,I;{\widehat{\theta }}_{in}^k,{\widehat{\theta }}_{out}^k\}\right)\hfill \end{array}\phantom{\}}$

–Minimization w.r.t. w:

$\{\begin{array}{c}\text{Estimate transformation}T\text{by minimizing}{\Vert {\widehat{\varphi }}^k\circ T-{\psi }_0\Vert }_2^2\hfill \\ \text{Pushforward to shape dictionary coordinates}{\widehat{\varphi }}^k\leftarrow {\widehat{\varphi }}^k\circ T\hfill \\ \text{Minimization}\mathrm{w}.\mathrm{r}.\mathrm{t}.\mathbf{w}\text{using}\mathrm{ADMM}\text{algorithm}\left[24\right]:\hfill \\ {\widehat{\mathbf{w}}}^{k+1}\leftarrow \mathrm{ADMM}(\frac{\lambda }{2}{\Vert {\widehat{\varphi }}^k-D\mathbf{w})\Vert }_2^2+\frac{\gamma }{2}{\Vert C\mathbf{w}\Vert }_1)\hfill \\ \text{Pullback shape prior to current image coordinate}{\psi }^{k+1}\leftarrow D{\widehat{\mathbf{w}}}^{k+1}\circ T^{-1}\hfill \end{array}\phantom{\}}$

–Variational descent w.r.t. ϕ:

$\{\begin{array}{c}{\widehat{\varphi }}_1^{k+1}\leftarrow {\widehat{\varphi }}_{\mathrm{inner}\_\mathrm{iter}\_\mathrm{num}}^k\hfill \\ \text{for}i=1:\mathrm{inner}\_\mathrm{iter}\_\mathrm{num}\hfill \\ \begin{array}{c}\hfill {\widehat{\varphi }}_{i+1}^{k+1}={\widehat{\varphi }}_i^{k+1}+\Delta t\{-{\delta }_{∊}\left({\widehat{\varphi }}_i^{k+1}\right)log{\sum }_{j=1}^{n_{in}}\frac{{\widehat{w}}_{in,j}^{k+1}}{\sqrt{2\pi }{\widehat{\sigma }}_{in,j}^{k+1}}\exp (-\frac{{(I\left(x\right)-{\widehat{\mu }}_{in,j}^{k+1})}^2}{2{\left({\widehat{\sigma }}_{in,j}^{k+1}\right)}^2})\hfill \\ \hfill +{\delta }_{∊}\left({\widehat{\varphi }}_i^{k+1}\right)log{\sum }_{j=1}^{n_{out}}\frac{{\widehat{w}}_{out,j}^{k+1}}{\sqrt{2\pi }{\widehat{\sigma }}_{out,j}^{k+1}}\exp (-\frac{{(I\left(x\right)-{\widehat{\mu }}_{out,j}^{k+1})}^2}{2{\left({\widehat{\sigma }}_{out,j}^{k+1}\right)}^2})\hfill \\ \hfill -\beta {\delta }_{∊}\left({\widehat{\varphi }}_i^{k+1}\right)\nabla \cdot \left(g\frac{\nabla \varphi }{\mid \nabla \varphi \mid }\right)+\lambda (-{\widehat{\varphi }}_i^{k+1}+{\psi }^{k+1})\hfill \\ \hfill -\gamma \left({\sum }_{l=1}^m{\widehat{\mathbf{w}}}^{k+1}\left(l\right)({\widehat{\varphi }}_i^{k+1}-{\psi }^{k+1})\right)\}\hfill \end{array}\hfill \\ \text{end for}\hfill \end{array}\phantom{\}}$

end while