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Multiresolution analysis

Module by: Veronique Delouille. E-mail the author

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.

Definition

Definition:

( Multiresolution analysis) A multiresolution analysis of L2(IR)L2(IR) is defined as a sequence of closed subspaces VjL2(IR),jZZVjL2(IR),jZZ with the following properties:
  1. ... V - 1 V 0 V 1 ... ... V - 1 V 0 V 1 ...
    (1)
  2. The spaces VjVj satisfy
    jZZVjisdenseinL2(IR)andjZZVj={0}jZZVjisdenseinL2(IR)andjZZVj={0}
    (2)
  3. If f(x)V0,f(2jx)Vj.f(x)V0,f(2jx)Vj. This property means that all the spaces VjVj are scaled versions of the central space V0.V0.
  4. If fV0,f(.-k)V0,kZZ.fV0,f(.-k)V0,kZZ. That is, V0V0(and hence all the VjVj) is invariant under translation.
  5. There exists ϕV0ϕV0 such that {ϕ0,n;nZZ}{ϕ0,n;nZZ} is an orthonormal basis in V0.V0.

Condition 5 in Section 1 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 VjVj 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 ϕ ( x ) V 0
    (3)
  2. {ϕ(.-k),kZZ}{ϕ(.-k),kZZ} is an orthonormal basis for V0V0
  3. ϕ(2jx)Vjϕ(2jx)Vj,

we obtain that, for fixed jj, {ϕj,k(x)=2j/2ϕ(2jx-k),kZZ}{ϕj,k(x)=2j/2ϕ(2jx-k),kZZ} is an orthonormal basis for VjVj.

Since ϕV0V1,ϕV0V1, we can express ϕϕ as a linear combination of {ϕ1,k}:{ϕ1,k}:

ϕ ( x ) = k h k ϕ 1 , k ( x ) = 2 k h k ϕ ( 2 x - k ) . ϕ ( x ) = k h k ϕ 1 , k ( x ) = 2 k h k ϕ ( 2 x - k ) .
(4)

Equation 4 is called the refinement equation, or the two scales difference equation. The function ϕ(x)ϕ(x) is called the scaling function. Under very general condition, ϕϕ is uniquely defined by its refinement equation and the normalisation

- + ϕ ( x ) d x = 1 . - + ϕ ( x ) d x = 1 .
(5)

The spaces VjVj 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 VjVj is dense in L2(IR),L2(IR), we are guaranteed that any given function of L2L2 can be approximated arbitrarily close by such projections, i.e.:

lim j P j f = f , lim j P j f = f ,
(6)

for all ff in L2.L2. Note that the orthogonal projection of ff onto VjVj can be written as:

P j f = k Z Z α k ϕ j k . P j f = k Z Z α k ϕ j k .
(7)

where αk=<f,ϕj,k>.αk=<f,ϕj,k>.

Example

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 ϕ ( x ) = I 1 [ 0 , 1 ] = 1 if 0 x 1 0 otherwise
(8)

Hence we have that

ϕ ( 2 x ) = 1 if 0 x 1 / 2 0 otherwise ϕ ( 2 x ) = 1 if 0 x 1 / 2 0 otherwise
(9)

and

ϕ ( 2 x - 1 ) = 1 if 1 / 2 x 1 0 otherwise ϕ ( 2 x - 1 ) = 1 if 1 / 2 x 1 0 otherwise
(10)

The function ϕϕ generates, by translation and scaling, a multiresolution analysis for the spaces VjVj defined by:

V j = { f L 2 ( I R ) ; k Z Z , f | [ 2 j k , 2 j ( k + 1 ) [ = constant } V j = { f L 2 ( I R ) ; k Z Z , f | [ 2 j k , 2 j ( k + 1 ) [ = constant }
(11)

The wavelet function and the detail spaces Wj

The detail space Wj

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

V j + 1 = V j W j V j + 1 = V j W j
(12)

This space WjWj answers our question: it contains the “detail” information needed to go from an approximation at resolution jj to an approximation at resolution j+1.j+1. Consequently, by using recursively the Equation 12, we have:

j Z Z W j = L 2 ( I R ) . j Z Z W j = L 2 ( I R ) .
(13)

The main interest of MRA lies in the fact that, whenever a collection of closed subspaces satisfies the conditions in Definition 1, there exists an orthonormal wavelet basis, noted {ψj,k,kZZ}{ψj,k,kZZ} of Wj.Wj. Hence, we can say in very general terms that the projection of a function ff onto Vj+1Vj+1 can be decomposed as

P j + 1 f = P j f + Q j f , P j + 1 f = P j f + Q j f ,
(14)

where PjPj is the projection operator onto Vj,Vj, and QjQj is the projection operator onto Wj.Wj. 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 WjWj are defined as the orthogonal complement of VjVj in Vj+1.Vj+1. Consequently, the spaces Wj,Wj, with jZZjZZ are all mutually orthogonal, the projections PjPj and QjQj are orthogonal, so that the expansion

f ( x ) = j Q j f ( x ) f ( x ) = j Q j f ( x )
(15)

is orthogonal. Moreover, as QjQj is orthogonal, the projection onto WjWj can explicitely be written as:

Q j f ( x ) = k β j k ψ j k ( x ) , Q j f ( x ) = k β j k ψ j k ( x ) ,
(16)

with

β j k = < f , ψ j k > . β j k = < f , ψ j k > .
(17)

A sufficient condition for a MRA to be orthogonal is:

W 0 V 0 , W 0 V 0 ,
(18)

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

An orthogonal wavelet is a function ψψ such that the collection of functions {ψ(x-l)|lZZ}{ψ(x-l)|lZZ} is an orthonormal basis of W0.W0. This is the case if <ψ,ψ(.-l)>=δl,0.<ψ,ψ(.-l)>=δl,0.

In the following, we consider only orthogonal wavelets. We now outline how to construct a wavelet function ψ(x)ψ(x) starting from ϕ(x),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) {ϕj,k,kZZ}{ϕj,k,kZZ} for VjVj and we want to construct ψjkψjk such that

  • ψjk,kZZψjk,kZZ form an ONB for WjWj
  • V j W j , i.e. < ϕ j k , ψ j k ' > = 0 k , k ' V j W j , i.e. < ϕ j k , ψ j k ' > = 0 k , k '
    (19)
  • W j W j ' for j j ' . W j W j ' for j j ' .
    (20)

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

  1. Since ϕV0V1,ϕV0V1, and the ϕ1,kϕ1,k are an ONB in V1,V1, we have:
    ϕ(x)=khkϕ1,k,hk=<ϕ,ϕ1,k>,kZZ|hk|2=1.ϕ(x)=khkϕ1,k,hk=<ϕ,ϕ1,k>,kZZ|hk|2=1.
    (21)
    (refinement equation)
  2. δk,0=ϕ(x)ϕ(x-k)¯dxδk,0=ϕ(x)ϕ(x-k)¯dx
    (22)
    (orthonormality of ϕ(.-k)ϕ(.-k))
  3. Let us now characterize W0:fW0W0:fW0 is equivalent to fV1fV1 and fV0.fV0. Since fV1,fV1, we have:
    f=nfnϕ1,n,withfn=<f,ϕ1,n>.f=nfnϕ1,n,withfn=<f,ϕ1,n>.
    (23)
    The constraint fV0fV0 is implied by fϕ0,kfϕ0,k for all k.k.
  4. Taking the general form of fW0,fW0, we can deduce a candidate for our wavelet. We then need to verify that the ψ0,kψ0,k are indeed an ONB of W0.W0.

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 [1], chapter 5). Let us now state the result of these manipulations.

Theorem:

(Daubechies, chap 5) If a ladder of closed subspaces (Vj)jZZ(Vj)jZZ in L2(IR)L2(IR) satisfies the conditions of the Definition 1, then there exists an associated orthonormal wavelet basis {ψj,k;j,kZZ}{ψj,k;j,kZZ} for L2(IR)L2(IR) such that:
P j + 1 = P j + k < . , ψ j , k > ψ j , k . P j + 1 = P j + k < . , ψ j , k > ψ j , k .
(24)

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

ψ ( x ) = ( 2 ) n ( - 1 ) n h 1 - n ϕ ( 2 x - n ) ψ ( x ) = ( 2 ) n ( - 1 ) n h 1 - n ϕ ( 2 x - n )
(25)

(with convergence of this serie is L2L2 sense).

Example(continued)

Let us see what the recipe of Theorem 1 gives for the Haar multiresolution analysis. In that case, ϕ(x)=1ϕ(x)=1 for 0x1,00x1,0 otherwise. Hence:

h n = 2 d x ϕ ( x ) ϕ ( 2 x - n ) = 1 / 2 if n = 0 , 1 0 otherwise h n = 2 d x ϕ ( x ) ϕ ( 2 x - n ) = 1 / 2 if n = 0 , 1 0 otherwise
(26)

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 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
(27)

Homogeneous and inhomogeneous representation of

Inhomogeneous representation

If we consider a first (coarse) approximation of fV0,fV0, and then “refine” this approximation with detail spaces Wj,Wj, the decomposition of ff 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 ) , 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 ) ,
(28)

where αk=<f,ϕ0,k>αk=<f,ϕ0,k> and βj,k=<f,ψj,k>.βj,k=<f,ψj,k>. In this case, we talk about inhomogeneous representation of f.f.

Homogeneous representation

If we use the fact that

j Z Z W j is dense in L 2 ( I R ) , j Z Z W j is dense in L 2 ( I R ) ,
(29)

we can decompose ff as a linear combination of functions ψj,kψj,k only:

f ( x ) = j = - + k β j , k ψ j , k ( x ) . f ( x ) = j = - + k β j , k ψ j , k ( x ) .
(30)

We then talk about homogeneous representation of ff.

Properties of the homogeneous representation

  • Each coefficient βj,kβj,k in Equation 30 depends only locally on ff because
    βj,k=f(x)ψj,k(x)dx,βj,k=f(x)ψj,k(x)dx,
    (31)
    and the wavelet ψj,k(x)ψj,k(x) has (essentially) bounded support.
  • βj,kβj,k gives information on scale 2-j,2-j, near position 2-jk2-jk
  • a discontinuity in ff only affects a small proportion of coefficients– a fixed number at each frequency level.

References

  1. C.K. Chui. (1992). An Introduction to Wavelets. San Diego, CA: American Press.
  2. I. Daubechies. (1992). Ten Lectures on wavelets. Philadelphia: Society for Industrial and Applied Mathematics.
  3. W. Härdle, G. Kerkyacharian, D. Picard, and A. Tsybekov. (1998). Wavelets, Approximation, and Statistical Applications. Lectures notes in Statistics.
  4. S. Mallat. (1989). A theory for multiresolution signal decomposition: the wavelet representation. I.E.E.E Trans. on pattern analysis and machine intelligence, 11(7), 674-693.
  5. Y. Meyer. (1990). Ondelettes et Opérateurs, I: Ondelettes, II:. Operateurs de Calderón-Zygmund.

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