Data: A fixed image F and moving image M.

Result: Transformation Γ so that M ∘ Γ is “close” to F.

Set ϒ0 = identity transformation (or some a-priori transformation, e.g., from a previous registration)

repeat

Step 1. Given ϒ(t),

Minimize the first two terms of Eq. (3)

$$u^{(t)}=\underset{u}{\text{argmin}}{\Vert {\sum }^{-1}\left(F-M\circ \{{\Upsilon }^{(t)}\circ u\}\right)\Vert }^2+\frac{1}{{\sigma }_x^2}\text{dist}\left({\Upsilon }^{(t)},\{{\Upsilon }^{(t)}\circ u\}\right),$$ (1)

where u is any admissible transformation. Compute Γ(t) = ϒ(t) ∘ u(t).

Step 2. Given Γ(t),

Minimize the last two terms of Eq. (3):

$${\Upsilon }^{(t+1)}=\underset{\Upsilon }{\text{argmin}}\frac{1}{{\sigma }_x^2}\text{dist}(\Upsilon ,{\Gamma }^{(t)})+\frac{1}{{\sigma }_T^2}\text{Reg}(\Upsilon ).$$ (2)

until convergence;