above, the rotation is modeled as a rotation matrix with matrix level constraints. Thus it is unclear about the incorporation of inertial measurements. In addition, there are also specialized globally optimal algorithms focusing on one class of problem. In [@yang2015go][@liu2018efficient], pairs of features are used to decouple the 3D-3D registration. In [@yang2019polynomial], TEASER is proposed to decoupled scaled 3D-3D registration, achieving a fast three-stage optimization. These works show that it is possible to have superior performance with *specialized* algorithms rather than only the *general-purpose* framework, even also accelerated.

In this paper, we follow the idea of specialized solver to bridge the gap of globally optimal deterministic solution for visual inertial localization, which is a robust 3D-2D pose estimation problem with inertial measurements. To the best of our knowledge, this is the first work to study this problem in the context of global optimality. We expect this solution to be accurate and efficient.

Decoupling Translation and Rotation
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The underlying problem of visual inertial localization is the pose estimation from 3D-2D correspondences with outliers. Formally, given a set $\mathfrak{P}$ consisting of correspondences between 3D global points $p_i \in \mathbb{R}^3$ and 2D visual points $u_i \in \mathbb{R}^2$, they satisfy $$\label{3d2d}
  u_i = 