Analysis of deformational transformations with spatio-temporal continuous wavelet transforms

This paper deals with the estimation of deformational parameters in discrete spatio-temporal signals. The parameters of concern correspond to time-varying scales. As such they can be the coefficients of either a Taylor expansion of the scale or a given deformational transformation. At first sight there are just a few deformational transformations that provide continuous wavelet transforms. The approach presented in this paper associates deformational transformations to motion transformations taking place in higher dimensional spaces and projected on the sensor plane. Then finding continuous wavelet transforms becomes much easier since numerous continuous wavelet transforms have already been defined for motion analysis. It is also known that spatio-temporal continuous wavelet transforms provide minimum-mean-squared-error estimates of motion parameters. Any deformational transformation of features embedded in a spatio-temporal signal may always be related to the projection on the sensor plane of the motion of a rigid object taking place in a higher dimensional space. This reasoning applies conversely. The associated rigid motion may be actual or virtual may take place either on a flat space or on a curved space immersed in higher dimensions. Continuous wavelet transforms for the estimation of deformational parameters may be then deduced from those already existing in motion analysis.