  This paper focuses on designing a consistent and efficient filter for
map-based visual-inertial localization. First, we propose a new Lie group with
its algebra, based on which a novel invariant extended Kalman filter (invariant
EKF) is designed. We theoretically prove that, when we do not consider the
uncertainty of the map information, the proposed invariant EKF can naturally
maintain the correct observability properties of the system. To consider the
uncertainty of the map information, we introduce a Schmidt filter. With the
Schmidt filter, the uncertainty of the map information can be taken into
consideration to avoid over-confident estimation while the computation cost
only increases linearly with the size of the map keyframes. In addition, we
introduce an easily implemented observability-constrained technique because
directly combining the invariant EKF with the Schmidt filter cannot maintain
the correct observability properties of the system that considers the
uncertainty of the map information. Finally, we validate our proposed system's
high consistency, accuracy, and efficiency via extensive simulations and
real-world experiments.
