Second generation wavelet transform
Encyclopedia
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 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 (so to speak first generation) transforms such as the DWT
and CWT
).
The idea of moving away from the Fourier
domain was introduced independently by David Donoho
and Harten in the early 1990s.
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 (known as the updating operator) 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 (interpolating and updating) are repeated several times before the result is produced.
The idea can be expanded (as used in the DWT) to create a filter bank
with a number of levels.
The variable tree used in wavelet packet decomposition
can also be used.
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.
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
, the second generation wavelet transform (SGWT) is a wavelet
Wavelet
A wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully crafted to have...
transform where the filters
Filter (signal processing)
In signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...
(or even the represented wavelets) are not designed explicitly, but the transform consists of the application of the Lifting scheme
Lifting scheme
The lifting scheme is a technique for both designing wavelets and performing the discrete wavelet transform.Actually it is worthwhile to merge these steps and design the wavelet filters while performing the wavelet transform....
.
Actually, the sequence of lifting steps could be converted to a regular discrete wavelet transform
Discrete wavelet transform
In numerical analysis and functional analysis, a discrete wavelet transform is any wavelet transform for which the wavelets are discretely sampled...
, 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
Frequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....
, as they are usually in the classical (so to speak first generation) transforms such as the DWT
Discrete wavelet transform
In numerical analysis and functional analysis, a discrete wavelet transform is any wavelet transform for which the wavelets are discretely sampled...
and CWT
Continuous wavelet transform
A continuous wavelet transform is used to divide a continuous-time function into wavelets. Unlike Fourier transform, the continuous wavelet transform possesses the ability to construct a time-frequency representation of a signal that offers very good time and frequency localization...
).
The idea of moving away from the Fourier
Fourier
Fourier most commonly refers to Joseph Fourier , French mathematician and physicist, or the mathematics, physics, and engineering terms named in his honor for his work on the concepts underlying them:In mathematics:...
domain was introduced independently by David Donoho
David Donoho
David Leigh Donoho, born on March 5, 1957 in Los Angeles, is a professor of statistics at Stanford University, where he is also the Anne T. and Robert M. Bass Professor in the Humanities and Sciences...
and Harten in the early 1990s.
Calculating transform
The input signal is split into odd and even samples using shifting and downsamplingDownsampling
In signal processing, downsampling is the process of reducing the sampling rate of a signal. This is usually done to reduce the data rate or the size of the data....
. The detail coefficients
are then interpolated using the values of and the prediction operator on the even values:The next stage (known as the updating operator) 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 (interpolating and updating) are repeated several times before the result is produced.
The idea can be expanded (as used in the DWT) to create a filter bank
Filter bank
In signal processing, a filter bank is an array of band-pass filters that separates the input signal into multiple components, each one carrying a single frequency subband of the original signal. One application of a filter bank is a graphic equalizer, which can attenuate the components...
with a number of levels.
The variable tree used in wavelet packet decomposition
Wavelet packet decomposition
Wavelet packet decomposition is a wavelet transform where the signal is passed through more filters than the discrete wavelet transform ....
can also be used.
Advantages
The SGWT has a number of advantages over the classical wavelet transform in that it is quicker to compute (by a factor of 2) and it can be used to generate a multiresolution analysisMultiresolution analysis
A multiresolution analysis or multiscale approximation is the design method of most of the practically relevant discrete wavelet transforms and the justification for the algorithm of the fast wavelet transform...
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.