for details), then we have $$\begin{aligned}
\label{costpsys}
  &\max_{t,\{z_{ij}\}}\sum z_{ij} \\
  &s.t.~~~~ z_{ij} |\hat{t}_{ij}-t| \leq n_{ij},~~i \in \mathfrak{P},j \in \mathfrak{P} \cup \mathfrak{L}\end{aligned}$$ Now we find that the constraints are decoupled for each dimension of $t$. Set the $x$-dimension as example, we have $$\begin{aligned}
\label{cost1d}
  &\max_{t_x,\{z_{ij}\}}\sum z_{ij} \\
  &s.t.~~~~ z_{ij} |\hat{t}_{ij,x}-t_x| \leq n_{ij,x},~~i \in \mathfrak{P},j \in \mathfrak{P} \cup \mathfrak{L}\end{aligned}$$ arriving at the resultant three dimension-wise linear constrained consensus maximization problems.

**Dimension-wise voting algorithm.** We use a voting algorithm to solve the problem. We first specify the noise bound $n_{ij,x}$ in (\[cost1d\]). Given the noise bound $n_{i}$ in (\[costlsys\]), we have the noise bound for $t$ following the techniques in [@mccormick1976computability] [@sherali1992new] as $$\label{tbound}
  \underline{t}_{ij} \leq \hat{t}_{ij} \leq \overline{t}_{ij}$$ The details can be found in Appendix.

Still taking $x$-dimension as example, each estimated $\hat{t}_{ij,x}$ defines an interval $[\underline{t}_{ij,x},\overline{t}_{ij,x}]$. If the real $t_x$ lies in this interval, then the real inlier set contains the two correspondences deriving $\hat{t}_{ij,x}$. According to [@yang2019polynomial], the insight is that the inlier set only changes its membership when real $t_x$ enters a new interval. Besides, given $K$ estimations, the *maximum number of possible consensus sets, i.e. the cardinality of the solution space, is $2K-1$*, where $K$ is in *quadratic* 