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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] .

Questions & Answers

are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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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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