into the problem for decoupling.

In this paper, we propose a deterministic visual inertial localization solution to achieve global convergence with much higher efficiency. The key idea is to divide $SE(3)$ search space into multiple 1-D search spaces. Specifically, inspired by the decoupling idea in [@yang2019polynomial], we build intermediate cost function for both point and line features, *translation invariant measurements* (TIMs), to decouple consensus maximization into two cascaded subproblems only related to rotation $SO(3)$ and translation $\mathbb{R}^3$ respectively. Based on TIMs, the globally optimal rotation is then searched by *1-dimensional BnB* in $[-\pi,\pi]$ with the aid of inertial measurements. For the translation, $\mathbb{R}^3$ search is replaced with three 1-dimensional $\mathbb{R}$ search for each axis using *prioritized progressive voting*. To the best of our knowledge, this is the first solver for visual inertial localization with deterministic global optimality. In summary, our contributions include

-   TIMs based formulation of visual inertial localization that decouples the problem and enables 1D BnB based global optimization of the rotation.

-   Prioritized progressive voting method that replaces $\mathbb{R}^3$ space search with three $\mathbb{R}$ search for global optimization of the translation.

-   Experiments on simulation and real-world cross-session datasets that validate the effectiveness and efficiency of the 