Deformable template representations of observed imagery,
model the variability of target pose via
the actions of the matrix Lie groups on rigid templates.
In this paper, we study the construction of minimum mean squared
error estimators on the special orthogonal group, $SO(n)$, for
pose estimation.
Due to the non-flat geometry of $SO(n)$,
the standard Bayesian formulation, of optimal estimators and their
characteristics,
requires modifications.
By utilizing Hilbert-Schmidt metric defined on $GL(n)$,
a larger group containing $SO(n)$,
a mean squared criterion is defined on $SO(n)$.
The Hilbert-Schmidt estimate
(HSE) is defined to be a minimum mean squared error estimator, restricted to $SO(n)$.
The expected error associated with HSE is shown to be a lower
bound, called Hilbert-Schmidt bound (HSB), on the
error incurred by any other estimator.
Analysis and
algorithms are presented for evaluating HSE and HSB in case
of both ground-based and air-borne targets.