Continuous and Discrete Wavelets for Acceleration Kinematics: A Motion Estimation Based on Lie Groups and Variational Principles

A group-theoretic framework is presented for acceleration transformations. The main purpose is to show the existence of families of spatio-temporal continuous wavelets, frames, and discrete wavelets related to these transformations. The main application of interest is the analysis of motion in space–time signals. The construction of this framework starts with the enumeration of Lie algebras as building blocks that provide all the observable kinematics that comply with the properties of the space–time under analysis. These classes of accelerated kinematics generalize the kinematics defined in the Galilei group. Exponentiation from Lie algebras defines locally compact exponential groups. Unitary, irreducible, and square-integrable group representations are thereafter derived in the function spaces and the signals to be analyzed, leading to the existence of continuous and discrete wavelets, frames all indexed with higher orders of temporal derivatives of the translational motion. Group representations and wavelets are tools that perform the local optimum estimation of pieces of trajectory. The adjunction of a variational principle of optimality is further necessary for building a global trajectory and for performing tracking. The Euler–Lagrange equation provides the motion equation of the moving system and the Noether's theorem derives the related constants of motion. Dynamic programming implements the algorithms for tracking and constructing the global trajectory. Finally, tight frames and bases enable signal decompositions along the trajectory of interest.