In the problem of
recognizing targets from their observed images, the estimation of target
orientations, as elements of the rotation group $SO(3)$, plays an
important role.
For $k$-objects the unknown parameter is an element of $SO(3)^k$.
Since $k$ may be unknown a-priori, the parameter space is
extended to ${\cal X} = \cup_{k=0}^{\infty} SO(3)^k$. In this
representation, both the target
orientations and their numbers have to be estimated simultaneously.
We present a Bayesian approach which builds a posterior probability $\mu$
measure on ${\cal X}$.
Then, utilizing a Markov jump-diffusion process $X(t)$,
we sample from this posterior to empirically generate the estimates.
The two components of $X(t)$, jumps and diffusions, are chosen in such a way that
the resulting Markov process has the desired ergodic property: averages
along its sample paths converge to the expectations under the posterior.
Proper choice of the diffusion parameters and the jump intensities
is demonstrated and the ergodic result associated with $X(t)$ is proven.
An example, involving the estimation of an airplane orientation, is used to illustrate
this jump-diffusion algorithm.