The lifting scheme is a technique for both designing
wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
s and performing the
discrete wavelet transform
In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal ...
(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 {{Short description, Type of wavelet transform
In signal processing, the second-generation wavelet transform (SGWT) is a wavelet transform where the filters (or even the represented wavelets) are not designed explicitly, but the transform consists o ...
. 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
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...
.
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.
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
In information technology, lossy compression or irreversible compression is the class of data compression methods that uses inexact approximations and partial data discarding to represent the content. These techniques are used to reduce data size ...
. 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
Cohen–Daubechies–Feauveau wavelets are a family of biorthogonal wavelets that was made popular by Ingrid Daubechies. These are not the same as the orthogonal Daubechies wavelets, and also not very similar in shape and properties. However, thei ...
, there is an explicit formula for their lifting steps.
Increasing vanishing moments, stability, and regularity
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. However, increasing the number of vanishing moments with a lifting also increases the wavelet support, which is an adverse effect that increases the number of large coefficients produced by isolated singularities. Each lifting step maintains the filter biorthogonality but provides no control on the Riesz bounds and thus on the stability of the resulting wavelet biorthogonal basis. When a basis is orthogonal then the dual basis is equal to the original basis. Having a dual basis that is similar to the original basis is, therefore, an indication of stability. As a result, stability is generally improved when dual wavelets have as much vanishing moments as original wavelets and a support of similar size. This is why a lifting procedure also increases the number of vanishing moments of dual wavelets. It can also improve the regularity of the dual wavelet. 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. This is the main weakness of this wavelet design procedure.
Generalized lifting
The generalized lifting scheme was developed by Joel Solé and Philippe Salembier and published in Solé's PhD dissertation. It is based on the classical lifting scheme and generalizes it by breaking out a restriction hidden in the scheme structure. The classical lifting scheme has three kinds of operations:
# A lazy wavelet transform splits signal