  Underwater cameras are typically placed behind glass windows to protect them
from the water. Spherical glass, a dome port, is well suited for high water
pressures at great depth, allows for a large field of view, and avoids
refraction if a pinhole camera is positioned exactly at the sphere's center.
Adjusting a real lens perfectly to the dome center is a challenging task, both
in terms of how to actually guide the centering process (e.g. visual servoing)
and how to measure the alignment quality, but also, how to mechanically perform
the alignment. Consequently, such systems are prone to being decentered by some
offset, leading to challenging refraction patterns at the sphere that
invalidate the pinhole camera model. We show that the overall camera system
becomes an axial camera, even for thick domes as used for deep sea exploration
and provide a non-iterative way to compute the center of refraction without
requiring knowledge of exact air, glass or water properties. We also analyze
the refractive geometry at the sphere, looking at effects such as forward- vs.
backward decentering, iso-refraction curves and obtain a 6th-degree polynomial
equation for forward projection of 3D points in thin domes. We then propose a
pure underwater calibration procedure to estimate the decentering from multiple
images. This estimate can either be used during adjustment to guide the
mechanical position of the lens, or can be considered in photogrammetric
underwater applications.
