Algorithm 1.

Alternating Optimization of Eq. (3) in the i-th iteration

Result: φ – the final deformation field and approximated solution of Eq. (3)

i = 0; ${\widehat{I}}_{\text{CT}}^0={\widehat{I}}_{\text{CT}};I_{\text{MR}}^0=I_{\text{MR}}$; φ = 0;

do{

i = i + 1; ${\phi }_1^i=\text{Register}({\widehat{I}}_{\text{CT}}^{i-1},I_{\text{CT}});{\phi }_2^i=\text{Register}(I_{\text{MR}}^{i=1},{\widehat{I}}_{\text{MR}})$;

${\phi }^i=\frac{1}{2}({\phi }_1^i+{\phi }_2^i)$; φ = φ∘φi; ${\widehat{I}}_{\text{CT}}^i=\mathcal{D}({\widehat{I}}_{\text{CT}},\phi );I_{\text{MR}}^i=\mathcal{D}(I_{\text{MR}},\phi )$;

} while (||φi||2 > ε);