  Constant biases associated to measured linear and angular velocities of a
moving object can be estimated from measurements of a static scene by embedded
brightness and depth sensors. We propose here a Lyapunov-based observer taking
advantage of the SO(3)-invariance of the partial differential equations
satisfied by the measured brightness and depth fields. The resulting asymptotic
observer is governed by a non-linear integro/partial differential system where
the two independent scalar variables indexing the pixels live on the unit
sphere of the 3D Euclidian space. The observer design and analysis are strongly
simplified by coordinate-free differential calculus on the unit sphere equipped
with its natural Riemannian structure. The observer convergence is investigated
under C^1 regularity assumptions on the object motion and its scene. It relies
on Ascoli-Arzela theorem and pre-compactness of the observer trajectories. It
is proved that the estimated biases converge towards the true ones, if and only
if, the scene admits no cylindrical symmetry. The observer design can be
adapted to realistic sensors where brightness and depth data are only available
on a subset of the unit sphere. Preliminary simulations with synthetic
brightness and depth images (corrupted by noise around 10%) indicate that such
Lyapunov-based observers should be robust and convergent for much weaker
regularity assumptions.
