Generalized Lifting

Generalized Lifting

Generalized lifting scheme was developed by Joel Solé and Philippe Salembier and published in Joel's PhD Thesis. It is based on classical lifting scheme and generalizes it breaking out a restriction hidden in the scheme structure. Classical lifting scheme has three kind of operations.

1. Lazy wavelet transform splits signal in two new signals: the odd samples signal denoted by and the even samples signal denoted by .
2. Prediction step its objective is compute a prediction for the odd samples, based on the even samples (or vice versa). This prediction is subtracted from the odd samples creating an error signal .
3. Update step This step recalibrates the low frequency branch with some of the energy removed during subsampling. In the case of classical Lifting, this is used in order to "prepare" the signal for the next prediction step. It uses the predicted odd samples to prepare the even ones (or vice versa). This update is subtracted from the even samples producing the signal denoted by .

The scheme is invertible due to the structure of itself. In the receiver the update step is computed first. Its result is added to the even samples. After that, it's possible to compute exactly the same prediction and add it to the odd samples. In order to recover the original signal, we have to invert the Lazy Wavelet Transform. Generalized lifting scheme has the same three kind of operations. However this scheme avoids the addition-subtraction restriction that offered Classical Lifting. That fact has some consequences. For example, the design of all steps must guarantee the scheme invertibility (not guaranteed if the addition-subtraction restriction is avoided).