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| In [[signal processing]], the '''second-generation wavelet transform (SGWT)''' is a [[wavelet]] transform where the [[filter (signal processing)|filters]] (or even the represented wavelets) are not designed explicitly, but the transform consists of the application of the [[Lifting scheme]].
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| 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.
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| 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 [[discrete wavelet transform|DWT]] and [[continuous wavelet transform|CWT]]).
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| The idea of moving away from the [[Fourier analysis|Fourier]] domain was introduced independently by [[David Donoho]] and [[Harten]] in the early 1990s.
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| == Calculating transform ==
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| The input signal <math>f</math> is split into odd <math>\gamma _1</math> and even <math>\lambda _1</math> samples using shifting and [[downsampling]]. The detail coefficients <math>\gamma _2</math> are then interpolated using the values of <math>\gamma _1</math> and the ''prediction operator'' on the even values:
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| :<math>\gamma _2 = \gamma _1 - P(\lambda _1 ) \, </math>
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| The next stage (known as the ''updating operator'') alters the approximation coefficients using the detailed ones:
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| :<math>\lambda _2 = \lambda _1 + U(\gamma _2 ) \, </math>
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| [[Image:Second generation wavelet transform.svg|center|500px|alt=Block diagram of the SGWT]]
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| The functions prediction operator <math>P</math> and updating operator <math>U</math>
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| effectively define the wavelet used for decomposition.
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| For certain wavelets the lifting steps (interpolating and updating) are repeated several times before the result is produced.
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| The idea can be expanded (as used in the DWT) to create a [[filter bank]] with a number of levels.
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| The variable tree used in [[wavelet packet decomposition]] can also be used.
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| == Advantages ==
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| 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 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.
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| The transform can be modified locally while preserving invertibility; it can even adapt to some extent to the transformed signal.
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| == References ==
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| * Wim Sweldens: [http://www.ima.umn.edu/industrial/97_98/sweldens/fourth.html Second-Generation Wavelets: Theory and Application]
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| [[Category:Wavelets]]
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