Transversal versus lifting approach to motion-compensated temporal discrete wavelet transform of image sequences: equivalence and tradeoffs

Lifting-based implementations of various discrete wavelet transforms applied in the temporal direction under motion compensation have recently become a very powerful tool in video compression research. We present in this paper a theoretical analysis of motion compensation in both transversal and lifted implementations of such transforms. We derive conditions for perfect reconstruction in the case of motion-compensated transversal discrete wavelet transform. We also derive conditions on motion transformation assuring that a motion-compensated lifting scheme is exactly equivalent to its transversal counterpart. In general, these conditions require that motion transformation allow composition and be invertible. Unfortunately, many motion models do not obey these properties, thus inducing subband decomposition errors (prior to compression). We propose an alternative approach to motion compensation in the case of Haar transform. This new approach poses no constraints on motion; motion-compensated lifted Haar transform exactly implements its transversal implementation, and the latter obeys perfect reconstruction, both regardless of motion transformation used. This new approach, however, does not extend to the 5/3 or any higher-order discrete wavelet transform.