Second-generation wavelet transform


In signal processing, the second-generation wavelet transform is a wavelet transform where the filters are not designed explicitly, but the transform consists of the application of the Lifting scheme.
Actually, the sequence of lifting steps could be converted to a regular discrete wavelet transform, but this is unnecessary because both design and application is made via the lifting scheme.
This means that they are not designed in the frequency domain, as they are usually in the classical.
The idea of moving away from the Fourier domain was introduced independently by David Donoho and Harten in the early 1990s.

Calculating transform

The input signal is split into odd and even samples using shifting and downsampling. The detail coefficients are then interpolated using the values of and the prediction operator on the even values:
The next stage alters the approximation coefficients using the detailed ones:
The functions prediction operator and updating operator
effectively define the wavelet used for decomposition.
For certain wavelets the lifting steps are repeated several times before the result is produced.
The idea can be expanded to create a filter bank with a number of levels.
The variable tree used in wavelet packet decomposition can also be used.

Advantages

The SGWT has a number of advantages over the classical wavelet transform in that it is quicker to compute and it can be used to generate a multiresolution analysis that does not fit a uniform grid. Using a priori information the grid can be designed to allow the best analysis of the signal to be made.
The transform can be modified locally while preserving invertibility; it can even adapt to some extent to the transformed signal.