Harmonic analysis associated with spatio-temporal transformations

Abstract

The paper presents new developments in harmonic analysis associated with the motion transformations embedded in digital signals. In this context, harmonic analysis provides motion analysis with a complete theoretical construction of perfectly matching concepts and a related toolbox leading to fast algorithms. This theory can be built from only two assumptions: an associative structure for the local motion transformations expressed as a Lie group and a principle of optimality for the global evolution expressed as a variational extremal. Motion analysis means not only detection, estimation, interpolation, and tracking but also propagator's motion-compensated filtering, signal decomposition, and selective reconstruction. The optimality principle defines the trajectory and provides the appropriate equations of motion, the selective tracking equations, the selective constants of motion to be tracked, and all the symmetries to be imposed on the system. The harmonic analysis provides new special functions, orthogonal bases, partial differential equations (PDE), ordinary differential equations (ODE) and integral transforms. The tools to be developed rely on group representations, continuous and discrete wavelets, the estimation theory (prediction, smoothing and interpolation) and filtering theory (Kalman filters, motion-based convolutions, integral transforms). All the algorithms are supported by fast and parallelizable implementations based on the fast Fourier transform (FFT) and dynamic programming.