Algorithm 1.

Outline of the proposed consistency-based registration rectification algorithm.

Data:$\mathcal{R}$, $\mathcal{X}$, $\lambda$, $\gamma$, $\sigma$

Result:${\mathcal{R}}^{\prime }$

$t\leftarrow 0$

${\mathcal{R}}^0\leftarrow \mathcal{R}$

whilenot-convergedand$t<t_{\max }$do

Compute ${\mathrm{\Psi }}^t$ using Eqs. (7)–(11)

Update ${\mathrm{\Psi }}^{\text{best}}$ and ${\mathcal{D}}^{\text{best}}$ according to Eq. (21)

Downsample ${\mathcal{R}}^t$ and ${\mathrm{\Psi }}^t$

forall theaxes$d$do

Form matrix $A_d$ and vector $b_d$ using Eqs. (17)–(22)

${\mathcal{R}}_d^t=\min {|A_{d}x-b_d|}^2$

end

${\mathcal{R}}^t\leftarrow {\cup }_d{\mathcal{R}}_d^t$

Upsample ${\mathcal{R}}^t$

$t\leftarrow t+1$

end

${\mathcal{R}}^{\prime }\leftarrow {\mathcal{R}}^t$