Importance of motion in motion-compensated temporal discrete wavelet transforms

Discrete wavelet transforms (DWTs) applied temporally under motion compensation (MC) have recently become a very powerful tool in video compression, especially when implemented through lifting. A recent theoretical analysis has established conditions for perfect reconstruction in the case of transversal MC-DWT, and also for the equivalence of lifted and transversal implementations of MC-DWT. For Haar MC-DWT these conditions state that motion must be invertible, while for higher-order transforms they state that motion composition must be a well-defined operator. Since many popular motion models do not obey these properties, thus inducing errors (prior to compression), it is important to understand what is the impact of motion non-invertibility or quasi-invertibility on the performance of video compression. In this paper, we present new experimental results of a study aiming at a quantitative evaluation of such impact in case of block-based motion. We propose a new metric to measure the degree with which two motion fields are not inverses of each other. Using, this metric we investigate several motion inversion schemes, from simple temporal sample-and-hold, through spatial nearest neighbor, to advanced spline-based inversion, and we compare compression performance of each method to that of independently-estimated forward and backward motion fields. We observe that compression performance monotonically improves with the reduction of the proposed motion inversion error, up to 1-1.5dB for the advanced spline-based inversion. We also generalize the problem of "unconnected" pixels by extending it to both update and prediction steps, as opposed to the update step only used in conventional methods. Initial tests show favorable results compared to previously reported techniques.