Uncertainty relations associated with motion analysis

The purpose of this work is to describe the uncertainty relations which rule the estimation of motion parameters embedded in spatio-temporal digital signals defined in L/sup 2/(R/sup 2//spl times/R, dx/spl I.oarr/dt) as well as the estimation of motion parameters taking place in the exterior scene R/sup 3//spl times/R. This problem is central for optimal motion tracking applications since it determines how to design the sequence of motion parameters to estimate. This analysis extends to motion on smooth manifolds (i.e. curved surfaces) and/or to sensor arrays which are deployed on smooth manifolds. Signal analysis means motion detection, estimation and selective reconstructions. It is performed with template functions or wavelets taken as cross-correlation functions. In this context, both kinematics and geometry are described by Lie algebras. The Lie algebras characterize all the actual and physical models that can be observed in the exterior scenes as well as in scenes captured by the sensors. The local transformations are structured in Lie groups that are computed by exponentiation from the corresponding algebras. Eventually, group representations and continuous wavelets are derived in the functional space of the signals. The important issue is that the Lie algebra contains all the information on the estimation uncertainties. Indeed, any pair of generators in the algebra that fails to commute generates a Heisenberg-type inequality. The harmonic analysis associated with the uncertainty principle may then be derived in the functional space of interest from the construction of group representations. The example of motion estimation on flat surface is considered. In this case, the admissibility of multi-dimensional square-integrable templates moving at constant velocity (modeled by the Galilei group) require that velocity and translation (position) generators do not commute. The generalization of this motion is thereafter studied on manifolds and generates highly non-commutative cases.