Irregular sampling problems and selective reconstructions associated with motion transformations

This paper introduces the irregular sampling problem associated with motion transformations embedded in image sequences. Moving patterns in image sequences undergo a sampling which is a function of the relative position of the object and the sampling grid. To solve this problem, it is effective to consider motion as a smooth invertible time-warping transformation. Important applications are related to this topic. Let us mention the focalization on selected moving areas characterized by a specific scale and a specific kinematic. Focalization and selective reconstruction can be performed either for analysis with interpolation, prediction, and de-noising or for coding with transmission of limited areas of interest. The Shannon sampling theorem and its generalizations as Kramer and Parzen theorems apply in this context with Clark's theorem. Clark's theorem shows that signals formed by warping band-limited signals admit formulae for reconstruction from samples. Furthermore, the warping operators that lift the pattern up to a trajectory are chosen as unitary irreducible and square-integrable group representations. These operators bring important tools to motion-selective analysis and reconstruction, namely continuous wavelets, frames, discrete wavelet transforms, and reproducing kernel subspaces. Two examples are treated with motion at constant translational velocity and angular velocity. It is shown that the analysis and reconstruction structures directly derived from motion-based groups are equivalent to warping the same structures from the usual affine multidimensional group defined for space-time transformations.