| In the context of rigid motion estimation, we present a Bayesian inference technique for parameterizations on curved manifolds such as matrix Lie groups. Representative patterns of complex dynamic systems are built via combinatorics of simpler constituent elements called generators. These representations, based on the Deformable Template theory, are composed of: (i) the pattern configurations formed by composition of pattern generators, and (ii) the continuous and discrete transformations on these patterns to model the scene variabilities. The continuous transformations are accomplished by the action of Lie groups (rotation and translation) on rigid surface templates of the targets; the discrete moves correspond to classical birth/death/relabel type moves in the configuration space. The configuration space C becomes a countable disjoint union of connected Lie groups, {dollar}{lcub}bf SE{rcub}(n)sp{lcub}l{rcub},{dollar} each with a different label {dollar}alpha{dollar} and dimensions d(l).; A Bayesian approach develops posterior probability models on the representation space and generates statistics under these distributions. A set of non-linear stochastic differential equations incorporating Newtonian mechanics are used to build prior distributions on flows in {dollar}{lcub}bf SE{rcub}(n), n = 2, 3.{dollar} For inducing measures under mappings from a probability space of driving forces to the parametric manifolds of target motion the base measures on these manifolds are derived. The statistical models on the multiple-sensor data collection, based on the physics of sensors, provide the data likelihood component completing the posterior distribution {dollar}mu{dollar} on {dollar}{lcub}cal C{rcub}.{dollar}; To generate minimum mean squared error (MMSE) statistics under {dollar}mu{dollar} we construct a Markov process {dollar}X(t)in{lcub}cal C{rcub}{dollar} to sample from {dollar}mu{dollar}, i.e. {dollar}mu{dollar} is the unique stationary measure of X(t). X(t) is chosen to satisfy jump-diffusion dynamics, i.e. the process jumps on random exponential times across subspaces SE({dollar}n)sp{lcub}l{rcub},{dollar} and between jumps it follows sample path continuous solutions of stochastic differential equations (SDEs), generating ergodic stochastic flows on group representations of system patterns. For X(t) satisfying the specific jump-diffusion dynamics it is be shown that {dollar}mu{dollar} is its stationary measure. In particular, the transition probabilities converge in the variation norm to {dollar}mu{dollar} and for a bounded measurable function {dollar}f,limsb{lcub}ttoinfty{rcub}{lcub}1over t{rcub}intsbsp{lcub}0{rcub}{lcub}t{rcub}f(X(s))dstointsb{lcub}cal C{rcub}f(c)dmu(c).{dollar}; Algorithms are presented for constructing these jump-diffusion processes in the context of rigid motion estimation and results are presented from a simulation of a multi-target multi-sensor environment. To analyze the performance of estimators taking values on matrix Lie groups, we have derived Hilbert-Schmidt lower bounds on the expected errors. The bounds are explicitly derived and evaluated for the estimating rigid object orientations being observed by multiple sensors. The variations of these bounds as functions of observation noise, sensor variability, and target pose, are analyzed. |
