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The Lifting and Dual Lifting Step
The Lifting and Dual Lifting Step
Wavelet Transform using Lifting
Wavelet Transform using Lifting


In Chapter: A multiresolution formulation of Wavelet Systems , we introduced the multiresolution analysis for the space of L 2 functions, where we have a set of nesting subspaces

{ 0 } V - 2 V - 1 V 0 V 1 V 2 L 2 ,

where each subspace is spanned by translations of scaled versions of a single scaling function φ ; e.g.,

V j = Span k { 2 j / 2 φ ( 2 j t - k ) } .

The direct difference between nesting subspaces are spanned by translations of asingle wavelet at the corresponding scale; e.g.,

W j = V j + 1 V j = Span k { 2 j / 2 ψ ( 2 j t - k ) } .

There are several limitations of this construction. For example, nontrivial orthogonal wavelets can not be symmetric. To avoid this problem, we generalizedthe basic construction, and introduced multiplicity- M ( M -band) scaling functions and wavelets in [link] , where the difference spaces are spanned by translationsof M - 1 wavelets. The scaling is in terms of the power of M ; i.e.,

φ j , k ( t ) = M j / 2 φ ( M j t - k ) .

In general, there are more degrees of freedom to design the M-band wavelets. However, the nested V spaces are still spanned by translations of a single scaling function.It is the multiwavelets that removes the above restriction, thus allowing multiple scaling functions to span the nested V spaces [link] , [link] , [link] . Although it is possible to construct M -band multiwavelets, here we only present results on the two-band case, as most of the researches in theliterature do.

Construction of two-band multiwavelets

Assume that V 0 is spanned by translations of R different scaling functions φ i ( t ) , i = 1 , ... , R . For a two-band system, we define the scaling and translation of these functions by

φ i , j , k ( t ) = 2 j / 2 φ i ( 2 j t - k ) .

The multiresolution formulation implies

V j = Span k { φ i , j , k ( t ) : i = 1 , ... , R } .

We next construct a vector scaling function by

Φ ( t ) = φ 1 ( t ) , ... , φ R ( t ) T .

Since V 0 V 1 , we have

Φ ( t ) = 2 n H ( n ) Φ ( 2 t - n )

where H ( k ) is a R × R matrix for each k Z . This is a matrix version of the scalar recursive equation [link] . The first and simplest multiscaling functions probably appear in [link] , and they are shown in [link] .

The Simplest Alpert Multiscaling Functions
The Simplest Alpert Multiscaling Functions

The first scaling function φ 1 ( t ) is nothing but the Haar scaling function,and it is the sum of two time-compressed and shifted versions of itself, as shown in [link] (a). The second scaling function can be easilydecomposed into linear combinations of time-compressed and shifted versions of the Haar scaling function and itself, as

φ 2 ( t ) = 3 2 φ 1 ( 2 t ) + 1 2 φ 2 ( 2 t ) - 3 2 φ 1 ( 2 t - 1 ) + 1 2 φ 2 ( 2 t - 1 ) .

This is shown in [link]

Multiwavelet Refinement Equation
Multiwavelet Refinement Equation  [link]

Putting the two scaling functions together, we have

φ 1 ( t ) φ 2 ( t ) = 1 0 3 / 2 1 / 2 φ 1 ( 2 t ) φ 2 ( 2 t ) + 1 0 - 3 / 2 1 / 2 φ 1 ( 2 t - 1 ) φ 2 ( 2 t - 1 ) .

Further assume R wavelets span the difference spaces; i.e.,

W j = V j + 1 V j = Span k { ψ i , j , k ( t ) : i = 1 , ... , R } .

Since W 0 V 1 for the stacked wavelets Ψ ( t ) there must exist a sequence of R × R matrices G ( k ) , such that

Ψ ( t ) = 2 k G ( k ) Φ ( 2 t - k )

These are vector versions of the two scale recursive equations [link] and [link] .

We can also define the discrete-time Fourier transform of H ( k ) and G ( k ) as

H ( ω ) = k H ( k ) e i ω k , G ( ω ) = k G ( k ) e i ω k .

Properties of multiwavelets

Approximation, regularity and smoothness

Recall from Chapter: Regularity, Moments, and Wavelet System Design that the key to regularity and smoothness is having enough number of zeros at π for H ( ω ) . For multiwavelets, it has been shown that polynomials can be exactly reproduced by translatesof Φ ( t ) if and only if H ( ω ) can be factored in special form [link] , [link] , [link] . The factorization is used to study the regularity and convergence of refinablefunction vectors [link] , and to construct multi-scaling functions with approximation and symmetry [link] . Approximation and smoothness of multiple refinable functions are also studied in [link] , [link] , [link] .

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Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
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