consensus maximization problem with TIM constraints only related to rotation compatible to the map having both point and line features as $$\begin{aligned}
\label{costrtim}
  &\max_{R(\alpha),\{z_{*}\}}\sum z_{*} \\
  &s.t.~~~~ z_{ij} |d_{p,ij}(\alpha)| \leq n_{ij},~~i,j \in \mathfrak{P}\\
  &~~~~~ z_k |d_{l,k}(\alpha)| \leq n_k,~~k \in \mathfrak{L}\end{aligned}$$

Two-stage consensus maximization solver
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With TIMs for both point and line correspondences, we decouple the original consensus maximization problem into rotation only problem, and translation only problem when the rotation is fixed. Accordingly, the proposed solver has two stages in cascade:

-   We estimate the rotation $\hat{R}$ by $R(\hat{\alpha})$ based on the TIMs in (\[costrtim\]). This estimator solves a 1D optimization problem and is described in Section \[er\].

-   We estimate the translation $\hat{t}$ based on the original consensus maximization in (\[costrt\]) where the rotation is assigned with $\hat{R}$. This estimator solves a $\mathbb{R}^3$ optimization problem and is described in Section \[et\].

Estimators of Rotation and Translation
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BnB based optimization for rotation {#er}
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We employ BnB strategy to solve problem (\[costrtim\]). The cost function in (\[costrtim\]) relates to $\alpha$ and $z_{*}$. But it is obvious that when $\alpha$ is determined, $\{z_{*}\}$ is simply derived by evaluating the constraints. So we denote the cost function as $E(\alpha)$ that 