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The scaling function and the subspaces

There are two ways to introduce wavelets: one is through the continuous wavelet transform, and the other is through multiresolution analysis (MRA), which is the presentation adopted here. Here we start by defining multiresolution analysis and thereafter we give one example of such MRA.


( Multiresolution analysis) A multiresolution analysis of L 2 ( I R ) is defined as a sequence of closed subspaces V j L 2 ( I R ) , j Z Z with the following properties:
  1. ... V - 1 V 0 V 1 ...
  2. The spaces V j satisfy
    j Z Z V j is dense in L 2 ( I R ) and j Z Z V j = { 0 }
  3. If f ( x ) V 0 , f ( 2 j x ) V j . This property means that all the spaces V j are scaled versions of the central space V 0 .
  4. If f V 0 , f ( . - k ) V 0 , k Z Z . That is, V 0 (and hence all the V j ) is invariant under translation.
  5. There exists ϕ V 0 such that { ϕ 0 , n ; n Z Z } is an orthonormal basis in V 0 .

Condition 5 in [link] seems to be quite contrived, but it can be relaxed (i.e.,instead of taking orthonormal basis, we can take Riesz basis). We will use the following terminology: a level of a multiresolution analysis is one of the V j subspaces and one level is coarser (respectively finer ) with respect to another whenever the index of the corresponding subspace is smaller (respectively bigger).

Consequence of the definition

Let us make a couple of simple observations concerning this definition. Combining the facts that

  1. ϕ ( x ) V 0
  2. { ϕ ( . - k ) , k Z Z } is an orthonormal basis for V 0
  3. ϕ ( 2 j x ) V j ,

we obtain that, for fixed j , { ϕ j , k ( x ) = 2 j / 2 ϕ ( 2 j x - k ) , k Z Z } is an orthonormal basis for V j .

Since ϕ V 0 V 1 , we can express ϕ as a linear combination of { ϕ 1 , k } :

ϕ ( x ) = k h k ϕ 1 , k ( x ) = 2 k h k ϕ ( 2 x - k ) .

[link] is called the refinement equation , or the two scales difference equation. The function ϕ ( x ) is called the scaling function . Under very general condition, ϕ is uniquely defined by its refinement equation and the normalisation

- + ϕ ( x ) d x = 1 .

The spaces V j will be used to approximate general functions (see an example below). This will be done by defining appropriate projections onto these spaces. Since the union of all the V j is dense in L 2 ( I R ) , we are guaranteed that any given function of L 2 can be approximated arbitrarily close by such projections, i.e.:

lim j P j f = f ,

for all f in L 2 . Note that the orthogonal projection of f onto V j can be written as:

P j f = k Z Z α k ϕ j k .

where α k = < f , ϕ j , k > .


The simplest example of a scaling function is given by the Haar function:

ϕ ( x ) = I 1 [ 0 , 1 ] = 1 if 0 x 1 0 otherwise

Hence we have that

ϕ ( 2 x ) = 1 if 0 x 1 / 2 0 otherwise


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

The function ϕ generates, by translation and scaling, a multiresolution analysis for the spaces V j defined by:

V j = { f L 2 ( I R ) ; k Z Z , f | [ 2 j k , 2 j ( k + 1 ) [ = constant }

The wavelet function and the detail spaces wj

The detail space wj

Rather than considering all our nested spaces V j , we would like to code only the information needed to go from V j to V j + 1 . Hence we define by W j the space complementing V j in V j + 1 :

V j + 1 = V j W j

This space W j answers our question: it contains the “detail” information needed to go from an approximation at resolution j to an approximation at resolution j + 1 . Consequently, by using recursively the [link] , we have:

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
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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
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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.
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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, 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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