is explained as the number of inliers given a yaw angle $\alpha$.

**Upper bound of cost function.** We then derive the upper bound of $E(\alpha)$ on the subset $\mathbb{A}$, denoted as $\overline{E}(\mathbb{A})$, where $\alpha \in \mathbb{A} \subseteq[-\pi,\pi]$. Recall (\[tim\]) and (\[timl\]), as the forms of point-TIM and line-TIM are the same, we denote them as $d(\alpha)$. The lower bound of $|d(\alpha)|$ on $\mathbb{A}$, denoted as $\underline{d}(\mathbb{A})$, is derived as $$\label{timlb}
  \underline{d}(\mathbb{A}) =  \min |a_1 \sin(\alpha + a_2) + d_3|$$ where the derivation of the coefficients are introduced in Appendix. Note that $\underline{d}(\mathbb{A})$ can be solved analytically without any iterations. Now we formulate a consensus maximization problem as $$\begin{aligned}
\label{costlb}
  &\max_{R(\alpha),\{z_{*}\},\alpha \in \mathbb{A}}\sum z_{*} \\
  &s.t.~~~~ z_{ij} \underline{d}_{p,ij}(\mathbb{A}) \leq n_{ij},~~i,j \in \mathfrak{P}\\
  &~~~~~ z_k \underline{d}_{l,k}(\mathbb{A}) \leq n_{k},~~k \in \mathfrak{L}\end{aligned}$$ where the problem is defined on $\mathbb{A}$, and the TIMs constraints are replaced with tight lower bounds, relaxing the constraints and yielding an optimistic estimation of $\hat{z}_{*}$. We then have $$\begin{aligned}
\label{costub}
  E(\alpha) \leq \overline{E}(\mathbb{A}) = \sum \hat{z}_{*},~~\alpha \in \mathbb{A}\end{aligned}$$ as a tight upper bound. The equality exists when all constraints give the same $\alpha$ with $d_{p,ij}(\alpha) = \underline{d}_{p,ij}(\mathbb{A})$ and $d_{l,k}(\alpha) = \underline{d}_{l,k}(\mathbb{A})$, which is only possible when noise is free.

**Accelerate 