Spatio-temporal wavelet transforms for digital signal analysis

The goal of this paper is to investigate spatio-temporal continuous wavelet transforms. A new wavelet family, called the Galilean wavelet, has been designed to tune to four main parameters, namely, scale, spatio-temporal position, spatial orientation, and velocity. The paper starts with the theory of motion-compensated wavelet filtering in the discrete realm of image processing. As a major difference from multi-dimensional homogeneous spaces, spatio-temporal signal involves motion that warps the signal along the trajectories. Modeling motion with 2-D affine transformations leads to spatio-temporal generalizations. Decomposition into elementary operators leads to developing transformation groups and exploiting the related representation theory. The construction of continuous spatio-temporal wavelets in R-n x R spaces is then handled with classical techniques of calculation. Close connections may then be established among all the spatio-temporal wavelet transforms through different sets of transformations. This approach generates a general framework for the study of future tools. Frames of wavelets are thereafter investigated to revisit discrete wavelet transforms in a more general way. Eventually, illustrations demonstrate the ability of Galilean wavelet transforms to analyze spatio-temporal signals. (C) 1997 Elsevier Science B.V.