The lifting scheme is a technique for both designing wavelets and performing the discrete wavelet transform (DWT). In an implementation, it is often worthwhile to merge these steps and design the wavelet filters while performing the wavelet transform. This is then called the second-generation wavelet transform. The technique was introduced by Wim Sweldens.
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. Treatment of signal boundaries is also simplified.
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. Then a series of convolution–accumulate operations across the divided signals is applied.
Basics
The simplest version of a forward wavelet transform expressed in the lifting scheme is shown in the figure above. means predict step, which will be considered in isolation. The predict step calculates the wavelet function in the wavelet transform. This is a high-pass filter. 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. In order to perform the DWT using the lifting scheme, the corresponding lifting and scaling steps must be derived from the biorthogonal wavelets. The analysis filters () of the particular wavelet are first written in polyphase matrix
where .
The polyphase matrix is a 2 × 2 matrix containing the analysis low-pass and high-pass filters, each split up into their even and odd polynomial coefficients and normalized. From here the matrix is factored into a series of 2 × 2 upper- and lower-triangular matrices, each with diagonal entries equal to 1. The upper-triangular matrices contain the coefficients for the predict steps, and the lower-triangular matrices contain the coefficients for the update steps. A matrix consisting of all zeros with the exception of the diagonal values may be extracted to derive the scaling-step coefficients. The polyphase matrix is factored into the form
where is the coefficient for the predict step, and is the coefficient for the update step.
An example of a more complicated extraction having multiple predict and update steps, as well as scaling steps, is shown below; is the coefficient for the first predict step, is the coefficient for the first update step, is the coefficient for the second predict step, is the coefficient for the second update step, is the odd-sample scaling coefficient, and is the even-sample scaling coefficient:
According to matrix theory, any matrix having polynomial entries and a determinant of 1 can be factored as described above. Therefore, every wavelet transform with finite filters can be decomposed into a series of lifting and scaling steps. Daubechies and Sweldens discuss lifting-step extraction in further detail.
CDF 9/7 filter
To perform the CDF 9/7 transform, a total of four lifting steps are required: two predict and two update steps. The lifting factorization leads to the following sequence of filtering steps.
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-reconstruction filter banks can be transformed into each other by a sequence of lifting steps. For a better understanding, if and are polyphase matrices with the same determinant, then the lifting sequence from to is the same as the one from the lazy polyphase matrix to .
Speedup
Speedup is by a factor of two. This is only possible because lifting is restricted to perfect-reconstruction filter banks. That is, lifting somehow squeezes out redundancies caused by perfect reconstruction.
The transformation can be performed immediately in the memory of the input data (in place, in situ) with only constant memory overhead.