Initialize partition of $[-\pi,\pi]$ into subsets $\{\mathbb{A}_{i}\}$.\
Initialize best estimation $E^{*}$, $\alpha^{*}$.\
Insert $\{\mathbb{A}_{i}\}$ into queue $q$.\

Prioritized progressive voting for translation {#et}
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When $R(\hat{\alpha})$ is estimated, the co-linear and co-planar constraints (\[linep\]) and (\[sp\]) are all linear constraints for $t$. Thus we can transform the consensus maximization problem with point and line constraints as $$\begin{aligned}
\label{costlsys}
  &\max_{t,\{z_i\}}\sum z_i \\
  &s.t.~~~~ z_i |A_i t+b_i| \leq n_{i},~~i \in \mathfrak{P} \cup \mathfrak{L}\end{aligned}$$ where $A_i \in \mathbb{R}^{1\times3}$ and $b_i \in \mathbb{R}$ are the coefficients for linear constraints derived from (\[linep\]) or (\[sp\]) with estimated $R(\hat{\alpha})$. However, this problem still has coupled constraints for $t$ so that $\mathbb{R}^3$ search is indispensable.

![The voting illustration of $\hat{t}_x$. Each $\hat{t}_{ij,x}$ derived by $i$-th and $j$-th correspondence votes for the interval if $[\omega_{i},\omega_{i+1}]\subseteq[\underline{t}_{ij,x},\overline{t}_{ij,x}]$, which means the corresponding consensus set contains i and j.[]{data-label="fig.vote"}](vote){width="50.00000%"}

**Decoupled linear constraints.** Note that for a point correspondence constraint (\[linep\]), we have two linear equations, while for a line correspondence constraint (\[sp\]), we have one. Therefore, given a pair of correspondences including at least one point correspondence, say the $i$-th point correspondence and the $j$-th point or line correspondence, it is sufficient to solve $\hat{t}_{ij}$ for this small linear system (see Appendix 