  Visual Inertial Odometry (VIO) is of great interest due the ubiquity of
devices equipped with both a monocular camera and Inertial Measurement Unit
(IMU). Methods based on the extended Kalman Filter remain popular in VIO due to
their low memory requirements, CPU usage, and processing time when compared to
optimisation-based methods. In this paper, we analyse the VIO problem from a
geometric perspective and propose a novel formulation on a smooth quotient
manifold where the equivalence relationship is the well-known invariance of VIO
to choice of reference frame. We propose a novel Lie group that acts
transitively on this manifold and is compatible with the visual measurements.
This structure allows for the application of Equivariant Filter (EqF) design
leading to a novel filter for the VIO problem. Combined with a very simple
vision processing front-end, the proposed filter demonstrates state-of-the-art
performance on the EuRoC dataset compared to other EKF-based VIO algorithms.
