correspondence derived the constraint are inliers.

### Line-TIM

Similar to a pair of point correspondences, given a set of line correspondences $\mathfrak{L}$, it is also possible to develop TIM. Given the end points of the image line segment $u_{k1}$ and $u_{k2}$, we have two un-normalized directions as (\[lineeq\]), denoted as $\tilde{u}_{k1}$ and $\tilde{u}_{k2}$.

Then following the fact that the point $p_k$ on the world line lies on the plane spanned by the rays from camera center along direction $\tilde{u}_{k1}$ and $\tilde{u}_{k2}$, we have $$\label{sp}
  (\tilde{u}_{k1} \times \tilde{u}_{k2})^T(Rp_k+t) = 0$$ which is a constraint for both rotation and translation. Since arbitrary number of points can be sampled from a line, we sample another point on the same world line to formulate the constraint as (\[sp\]). Then only one line correspondence can lead to line-TIM after proper substitution as $$\label{timl}
  d_l(\alpha) = d_{l,1} \sin \alpha + d_{l,2} \cos \alpha + d_{l,3}$$ where the line-TIM has the same form as point-TIM in (\[tim\]), but the coefficients are different. The derivation details are also presented in the Appendix.

**TIMs based rotation only problem.** Note that either (\[tim\]) or (\[timl\]) is only related to the yaw angle. By combining them together, we have a general 