\pi(Rp_i+t,K) + o_i + e_i$$ where $R \in SO(3)$ and $t \in \mathbb{R}^3$ is the camera pose to be estimated, $\pi$ is the camera projection function with known intrinsic parameters $K$, $|e_i| < n_i$ is assumed to be bounded random measurement noise, $o_i$ is zero for inlier while an arbitrary number for outlier. To deal with outliers, the robust pose estimation generally begins with consensus maximization problem as $$\begin{aligned}
\label{costrt}
  &\max_{R,t,\{z_i\}}\sum z_i \\
  &s.t.~~~~ z_i |u_i - \pi(Rp_i+t,K)| \leq n_i,~~i \in \mathfrak{P}\end{aligned}$$ where $z_i$ is binary, indicating whether $o_i$ is zero. To solve the problem in global, general BnB algorithms search in $SE(3)$, which is a coupled space of $SO(3)$ and $\mathbb{R}^3$. But this probably leads to exponential computational complexity in bad cases. For local techniques like RANSAC, inliers may be estimated conservatively, i.e. inliers regarded as outliers, especially when the noise is unavoidable.

Translation invariant measurements
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### Point-TIM

Inspired by the minimal solution in RANSAC, we develop an intermediate measurement which is invariant to the translation of the pose. Mathematically, given an image key point $u_i$, we have an un-normalized direction vector from the camera center as $$\label{lineeq}
  \tilde{u}_i \triangleq \left(
    \begin{array}{c}
      \tilde{u}_{i,x} \\
      \tilde{u}_{i,y} \\
      1 \\
    