Jump–diffusion Markov processes on orthogonal groups for object pose estimation

Abstract

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 X=⋃ k=0 ∞ SO(3) k . In this representation, both the target orientations and their numbers have to be estimated simultaneously. We present a Bayesian approach that builds a posterior probability μ measure on 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.