Lifting Scheme - Properties

Properties

• Perfect reconstruction
  • Every transform by the lifting scheme can be inverted.
  • Every perfect reconstruction filter bank can be decomposed into lifting steps by the Euclidean algorithm.
  • That is, "lifting decomposable filter bank" and "perfect reconstruction filter bank" denotes the same.
• Every two perfect reconstructable filter banks can be transformed into each other by a sequence of lifting steps. (If and are polyphase matrices with the same determinant, the lifting sequence from to, is the same as the one from the lazy polyphase matrix to .)
• Speedup by a factor of two. This is only possible because lifting is restricted to perfect reconstruction filterbanks. That is, lifting somehow squeezes out redundancies caused by perfect reconstructability.
• In place: The transformation can be performed immediately in the memory of the input data with only constant memory overhead.
• Non-linearities: The convolution operations can be replaced by any other operation. For perfect reconstruction only the invertibility of the addition operation is relevant. This way rounding errors in convolution can be tolerated and bit-exact reconstruction is possible. However the numeric stability may be reduced by the non-linearities. This must be respected if the transformed signal is processed like in lossy compression.

Although every reconstructable filter bank can be expressed in terms of lifting steps, a general description of the lifting steps is not obvious from a description of a wavelet family. However, for instance for simple cases of the Cohen-Daubechies-Feauveau wavelet, there is an explicit formula for their lifting steps. (See the respective article)