Generalized Lifting - Definition

Definition

Generalized lifting scheme is a dyadic transform that follows the next rules:

1. Computes a Lazy Wavelet Transform and splits even samples from odd samples.
2. Computes a Prediction Mapping. This step tries to predict odd samples taking into account the even ones (or vice versa). This a mapping from the space of the samples in to the space of the samples in . In this case the samples (from ) chosen to be the reference for are called the context. It could be expressed as:

3. Computes an Update Mapping. This step tries to update the even samples taking into account the odd predicted samples. It would be a kind of preparation for the next prediction step, if any. It could be expressed:

Obviously, these mapping cannot be any function. In order to guarantee the invertibility of the scheme itself, all mapping involved in the transform, must be invertible. In case that mappings arise and arrive on finite sets (discrete bounded value signals), this condition is equivalent to say that mappings are injective (one-to-one). Moreover, if mapping goes from one set to a set of the same cardinality, it should be bijective.

In the Generalized Lifting Scheme the addition/subtraction restriction is avoided by including this step in the mapping. In this way the Classical Lifting Scheme is generalized.