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A new motion parameter estimation algorithm based on the continuous wavelet transform
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This paper presents a group-theoretic approach for the analysis of rotational motion in image sequences. This method relies on Lie algebras, Lie groups and Lie group representations to provide not only the continuous wavelets but also the related tools of harmonic analysis. This approach can be referred to research works presented in J.S. Byrnes et al. (Wavelets and their Applications, Kluwer Academic Publishers, 1994) who strongly influenced this topic. For the purpose of modeling motion transformations, this paper introduces the concepts of Lie algebras and Lie groups as the actual mathematical foundations of all the observable kinematics embedded in spatio-temporal signals. Rotational motion analysis focuses on the estimation of angular velocity and angular accelerations embedded in image sequences. Rotational motion is usually carried on a trajectory, the complete problem at hand consists in estimating not only the angular velocity and its temporal derivatives but also the position, the translational velocity and its temporal derivatives along the carrier trajectory. The paper starts with the usual affine and Galilei groups and proceeds by successive extensions and sections to the rotational group. The theory of group representations is central to provide families of continuous wavelets, special functions, PDE's, ODE's and integral transforms as new mathematical tools of motion analysis in image sequence to perform optimal and selective detection, estimations, tracking, and reconstructions. This paper defines rotational wavelets and proposes a structured approach to perform estimation and tracking in image sequences which fits to Kalman filters. Simulations on real digital image sequences are also presented with tracking and estimation.