  This work presents a camera model for refractive media such as water and its
application in underwater visual-inertial odometry. The model is
self-calibrating in real-time and is free of known correspondences or
calibration targets. It is separable as a distortion model (dependent on
refractive index $n$ and radial pixel coordinate) and a virtual pinhole model
(as a function of $n$). We derive the self-calibration formulation leveraging
epipolar constraints to estimate the refractive index and subsequently correct
for distortion. Through experimental studies using an underwater robot
integrating cameras and inertial sensing, the model is validated regarding the
accurate estimation of the refractive index and its benefits for robust
odometry estimation in an extended envelope of conditions. Lastly, we show the
transition between media and the estimation of the varying refractive index
online, thus allowing computer vision tasks across refractive media.
