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j Z Z W j = L 2 ( I R ) .

The main interest of MRA lies in the fact that, whenever a collection of closed subspaces satisfies the conditions in [link] , there exists an orthonormal wavelet basis, noted { ψ j , k , k Z Z } of W j . Hence, we can say in very general terms that the projection of a function f onto V j + 1 can be decomposed as

P j + 1 f = P j f + Q j f ,

where P j is the projection operator onto V j , and Q j is the projection operator onto W j . Before proceeding further, let us state clearly what the term “orthogonal wavelet ” involves.

Recapitulation: orthogonal wavelet

We introduce above the class of orthogonal wavelets. Let us define this precisely.

An orthogonal wavelet basis is associated with an orthogonal multiresolution analysis that can be defined as follows. We talk about orthogonal MRA when the wavelet spaces W j are defined as the orthogonal complement of V j in V j + 1 . Consequently, the spaces W j , with j Z Z are all mutually orthogonal, the projections P j and Q j are orthogonal, so that the expansion

f ( x ) = j Q j f ( x )

is orthogonal. Moreover, as Q j is orthogonal, the projection onto W j can explicitely be written as:

Q j f ( x ) = k β j k ψ j k ( x ) ,

with

β j k = < f , ψ j k > .

A sufficient condition for a MRA to be orthogonal is:

W 0 V 0 ,

or < ψ , ϕ ( . - l ) > = 0 for l Z Z , since the other conditions simply follow from scaling.

An orthogonal wavelet is a function ψ such that the collection of functions { ψ ( x - l ) | l Z Z } is an orthonormal basis of W 0 . This is the case if < ψ , ψ ( . - l ) > = δ l , 0 .

In the following, we consider only orthogonal wavelets. We now outline how to construct a wavelet function ψ ( x ) starting from ϕ ( x ) , and thereafter we show what this construction gives with the Haar function.

How to construct ψ(χ) starting from ρ(χ)?

Suppose we have an orthonormal basis (ONB) { ϕ j , k , k Z Z } for V j and we want to construct ψ j k such that

  • ψ j k , k Z Z form an ONB for W j
  • V j W j , i.e. < ϕ j k , ψ j k ' > = 0 k , k '
  • W j W j ' for j j ' .

It is natural to use conditions given by the MRA aspect to obtain this. More specifically, the following relationships are used to characterize ψ :

  1. Since ϕ V 0 V 1 , and the ϕ 1 , k are an ONB in V 1 , we have:
    ϕ ( x ) = k h k ϕ 1 , k , h k = < ϕ , ϕ 1 , k > , k Z Z | h k | 2 = 1 .
    (refinement equation)
  2. δ k , 0 = ϕ ( x ) ϕ ( x - k ) ¯ d x
    (orthonormality of ϕ ( . - k ) )
  3. Let us now characterize W 0 : f W 0 is equivalent to f V 1 and f V 0 . Since f V 1 , we have:
    f = n f n ϕ 1 , n , with f n = < f , ϕ 1 , n > .
    The constraint f V 0 is implied by f ϕ 0 , k for all k .
  4. Taking the general form of f W 0 , we can deduce a candidate for our wavelet. We then need to verify that the ψ 0 , k are indeed an ONB of W 0 .

In fact, in our setting, the conditions given above can be re-written in the Fourier domain, where the manipulations become easier (for details,see [link] , chapter 5). Let us now state the result of these manipulations.

(Daubechies, chap 5) If a ladder of closed subspaces ( V j ) j Z Z in L 2 ( I R ) satisfies the conditions of the [link] , then there exists an associated orthonormal wavelet basis { ψ j , k ; j , k Z Z } for L 2 ( I R ) such that:
P j + 1 = P j + k < . , ψ j , k > ψ j , k .

One possibility for the construction of the wavelet ψ is to take

ψ ( x ) = ( 2 ) n ( - 1 ) n h 1 - n ϕ ( 2 x - n )

(with convergence of this serie is L 2 sense).

Example(continued)

Let us see what the recipe of [link] gives for the Haar multiresolution analysis. In that case, ϕ ( x ) = 1 for 0 x 1 , 0 otherwise. Hence:

h n = 2 d x ϕ ( x ) ϕ ( 2 x - n ) = 1 / 2 if n = 0 , 1 0 otherwise

Consequently,

ψ ( x ) = 2 h 1 ϕ ( 2 x ) - 2 h 0 ϕ ( 2 x - 1 ) = ϕ ( 2 x ) - ϕ ( 2 x - 1 ) = 1 if 0 x < 1 / 2 - 1 if 1 / 2 x 1 0 otherwise

Homogeneous and inhomogeneous representation of

Inhomogeneous representation

If we consider a first (coarse) approximation of f V 0 , and then “refine” this approximation with detail spaces W j , the decomposition of f can be written as:

f = P 0 f + j = 0 Q j f = k α k ϕ 0 , k ( x ) + j = 0 k β j , k ψ j , k ( x ) ,

where α k = < f , ϕ 0 , k > and β j , k = < f , ψ j , k > . In this case, we talk about inhomogeneous representation of f .

Homogeneous representation

If we use the fact that

j Z Z W j is dense in L 2 ( I R ) ,

we can decompose f as a linear combination of functions ψ j , k only:

f ( x ) = j = - + k β j , k ψ j , k ( x ) .

We then talk about homogeneous representation of f .

Properties of the homogeneous representation

  • Each coefficient β j , k in [link] depends only locally on f because
    β j , k = f ( x ) ψ j , k ( x ) d x ,
    and the wavelet ψ j , k ( x ) has (essentially) bounded support.
  • β j , k gives information on scale 2 - j , near position 2 - j k
  • a discontinuity in f only affects a small proportion of coefficients– a fixed number at each frequency level.

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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Source:  OpenStax, Multiresolution analysis, filterbank implementation, and function approximation using wavelets. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10568/1.2
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