Lifting scheme
[1] The lifting scheme factorizes any discrete wavelet transform with finite filters into a series of elementary convolution operators, so-called lifting steps, which reduces the number of arithmetic operations by nearly a factor two.
[2] The discrete wavelet transform applies several filters separately to the same signal.
In contrast to that, for the lifting scheme, the signal is divided like a zipper.
The simplest version of a forward wavelet transform expressed in the lifting scheme is shown in the figure above.
The update step calculates the scaling function, which results in a smoother version of the data.
As mentioned above, the lifting scheme is an alternative technique for performing the DWT using biorthogonal wavelets.
From here the matrix is factored into a series of 2 × 2 upper- and lower-triangular matrices, each with diagonal entries equal to 1.
A matrix consisting of all zeros with the exception of the diagonal values may be extracted to derive the scaling-step coefficients.
Therefore, every wavelet transform with finite filters can be decomposed into a series of lifting and scaling steps.
Every perfect-reconstruction filter bank can be decomposed into lifting steps by the Euclidean algorithm.
Every two perfect-reconstruction filter banks can be transformed into each other by a sequence of lifting steps.
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.
A lifting modifies biorthogonal filters in order to increase the number of vanishing moments of the resulting biorthogonal wavelets, and hopefully their stability and regularity.
Increasing the number of vanishing moments decreases the amplitude of wavelet coefficients in regions where the signal is regular, which produces a more sparse representation.
This is why a lifting procedure also increases the number of vanishing moments of dual wavelets.
A lifting design is computed by adjusting the number of vanishing moments.
The stability and regularity of the resulting biorthogonal wavelets are measured a posteriori, hoping for the best.
In order to recover the original signal, the lazy wavelet transform has to be inverted.
However, this scheme avoids the addition-subtraction restriction that offered classical lifting, which has some consequences.
Generalized lifting scheme is a dyadic transform that follows these rules: Obviously, these mappings cannot be any functions.
In case that mappings arise and arrive on finite sets (discrete bounded value signals), this condition is equivalent to saying that mappings are injective (one-to-one).
In the generalized lifting scheme the addition/subtraction restriction is avoided by including this step in the mapping.
The update-step design has not been considered as thoroughly, because it remains to be answered how exactly the update step is useful.
