Optimal group-theoretic methods for selective motion analysis: detection, estimation, filtering and reconstruction with continuous and discrete spatio-temporal wavelets

This work addresses the analysis of motion embedded in spatio-temporal digital signals as well as motion taking place in the outer space R³×R. Three categories of motion are considered and referred to as translational, rotational or deformational. In each category, motion parameters are defined from all the temporal derivatives i.e., position, velocity and accelerations. Motion analysis means not only detection, estimation, interpolation, and tracking but also motion-compensated filtering, signal decomposition, and selective reconstruction. In this context, we show how all motion models can be derived from Lie groups and how group representations define continuous wavelets in the functional space of the signals. Motion detection, estimation and interpolation are based on continuous wavelet transforms. Selective motion tracking is based on the adjunction of a variational principle of optimality. The optimality principle defines the trajectory or the geodesic and provides the appropriate PDE of wavelet motion, the tracking equation (ODE), the selective constants of motion to be tracked, and all the symmetries to be imposed on the system. The Green functions of these PDEs give rise to the converse operators i.e. wavelet propagators and kernels of integral equations. These integral equations have several applications. This work investigates in fact the harmonic analysis associated with motion groups which leads to special functions, spectral signatures, propagators and yields motion-based detection and velocity or motion-oriented filtering. This motion analysis fits to both deterministic and stochastic processes. Eventually, spatio-temporal discrete wavelets can be derived from their continuous cognates as the orthonormal bases that perform signal decompositions along the trajectory and achieve selective reconstructions of moving patterns of interest