Exponential Fourier Densities on $SO( 3 )$ and Optimal Estimation and Detection for Rotational Processes

Abstract

A new representation of a probability density function on the three dimensional rotation group, $SO( 3 )$ is presented, which generalizes the exponential Fourier densities on the circle. As in the circle case, this class of densities on $SO( 3 )$ is closed under the operation of taking conditional distributions. Several simple multistage estimation and detection models are considered. 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.An error criterion, which is compatible with a Riemannian metric, is introduced and discussed. The optimal orientation estimates with respect to this error criterion are derived for a given probability distribution, illustrating how the updated conditional densities can be used to recursively determine the optimal estimates on $SO( 3 )$.