w.r.t the number of correspondences. This complexity enables a voting algorithm for all $2K-1$ sets. By counting the unique correspondences of the votes in each set, we get the corresponding consensus set. Then the maximal consensus set can lead to an estimation of $\hat{t}_x$. An illustrative case is shown in Fig. \[fig.vote\] and the pseudo code is listed in Algorithm \[vt\] with $x$-dimension as example. For simplicity, we replace $\hat{t}_{ij,x}$ with $\hat{t}_{k,x}$ in the pseudo code. Following the similar idea in [@yang2019polynomial], by repeating the voting algorithm for three times, $\hat{t}$ is estimated as $[\hat{t}_x$, $\hat{t}_y$, $\hat{t}_z]^T$.

Initialize key-value map $S$.\
$\omega = sort([ \underline{t}_{1,x},\overline{t}_{1,x},\underline{t}_{2,x},\overline{t}_{2,x},..,\underline{t}_{K,x},\overline{t}_{K,x} ])$.\

**Prioritized progressive voting algorithm.** When the number of inliers is high, independent voting along three dimensions is possible. But when the number of inliers is low and outlier rate is high, independent dimension-wise voting may lead to failure. The reason is that, though it is almost impossible that there are more outliers than inliers having the similar $t$, *it is possible that there are more outliers than inliers having the similar* $t_x$. In such scenario, search along $x$-dimension leads to incorrect $\hat{t}_x$, which cannot be corrected in the successive voting along $y$ or $z$-dimension.

