Exponential Fourier densities on SO(3) and optimal estimation and detection for rotational processes

In this paper, we will present a new representation of a probability density function on the three dimensional rotation group, SO(3), which generalizes the exponential Fourier densities on the circle. As in the circle case, this class of densities on SO(3) is also closed under the operation of taking conditional distributions. Several simple multistage estimation and detection models will be considered in this paper. The closure property enables us to determine the sequential conditional densities by recursively updating a finite and fixed number of coefficients. It also enables us to express the likelihood ratio for signal detection explicitly in terms of the conditional densities.