<< Chapter < Page
  Wavelets and wavelet transforms     Page 13 / 28
Chapter >> Page >
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

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
how to synthesize TiO2 nanoparticles by chemical methods
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Wavelets and wavelet transforms' conversation and receive update notifications?