A natural way to introduce wavelets is through the multiresolution analysis. Given a function f∈L2(R)f∈L2(R), a multiresolution of L2(R)L2(R) will provide us with a sequence of spaces Vj,Vj+1,...Vj,Vj+1,... such that the projections of ff onto these spaces give finer and finer
approximations (as j→∞j→∞) of the function ff.
(Multiresolution analysis (MRA) in the first generation)
A multiresolution analysis of L2(R)L2(R) is defined as a sequence of closed subspaces Vj⊂L2(R),j∈ZVj⊂L2(R),j∈Z with the following properties:
-
...
⊂
V
-
1
⊂
V
0
⊂
V
1
⊂
...
...
⊂
V
-
1
⊂
V
0
⊂
V
1
⊂
...
(1)
- The spaces VjVj satisfy
⋃j∈ZVjisdenseinL2(R)and⋂j∈ZVj={0}.⋃j∈ZVjisdenseinL2(R)and⋂j∈ZVj={0}.(2)
- If f(x)∈V0,f(2jx)∈Vj,f(x)∈V0,f(2jx)∈Vj, i.e. the spaces VjVj are scaled versions of the central space V0.V0.
- If f∈V0,f(.-k)∈V0,k∈Zf∈V0,f(.-k)∈V0,k∈Z, that is,
V0V0 (and hence all the VjVj) is invariant under translation.
- There exists ϕ∈V0ϕ∈V0 such that {ϕ(x-k);k∈Z}{ϕ(x-k);k∈Z} is a Riesz basis in V0.V0.
We will call `level' of a MRA one of the subspaces VjVj.
From Definition 1, it follows that, for fixed jj, the set
{ϕjk(x)=2j/2ϕ(2jx-k);k∈Z}{ϕjk(x)=2j/2ϕ(2jx-k);k∈Z} of scaled and translated versions of ϕϕ is a Riesz basis for VjVj.
Since ϕ∈V0⊂V1ϕ∈V0⊂V1, we can express ϕϕ as a linear combination of {ϕ1,k}{ϕ1,k}:
ϕ
(
x
)
=
∑
k
∈
Z
h
k
ϕ
1
,
k
(
x
)
=
2
∑
k
∈
Z
h
k
ϕ
(
2
x
-
k
)
.
ϕ
(
x
)
=
∑
k
∈
Z
h
k
ϕ
1
,
k
(
x
)
=
2
∑
k
∈
Z
h
k
ϕ
(
2
x
-
k
)
.
(3)
Equation 3 is called the two-scale equation or refinement equation. It is a fundamental equation in MRA since it tells us how to go from a fine levelV1V1 to a coarser levelV0V0. The function ϕϕ is called the scaling function.
As said before, the spaces VjVj will be used to approximate general functions. This will be done by defining appropriate projections onto these spaces. Since the union of all the VjVj is dense in L2(R),L2(R), we are guaranteed that any given function of
L2(R)L2(R) can be approximated arbitrarily close by such projections.
As an example, define the space VjVj as
V
j
=
{
f
∈
L
2
(
R
)
;
∀
k
∈
Z
,
f
|
[
2
-
j
k
,
2
-
j
(
k
+
1
)
[
=
constant
}
V
j
=
{
f
∈
L
2
(
R
)
;
∀
k
∈
Z
,
f
|
[
2
-
j
k
,
2
-
j
(
k
+
1
)
[
=
constant
}
(4)
Then the scaling function ϕ(x)=1[0,1)(x)ϕ(x)=1[0,1)(x), called the Haar scaling function, generates by translation and dilatation a MRA for the sequence of spaces {Vj,j∈Z}{Vj,j∈Z} defined in Equation 4, see [4], [3].
Rather than considering all the nested spaces Vj,Vj, it would be more efficient to code only the information needed to go from VjVj to Vj+1.Vj+1. Hence we consider the space WjWj which complements VjVj in Vj+1Vj+1 :
V
j
+
1
=
V
j
⊕
W
j
.
V
j
+
1
=
V
j
⊕
W
j
.
(5)
The space WjWj is not necessarily orthogonal to VjVj, but it always
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 Equation 5, we have for any j0∈Zj0∈Z, the decomposition
L
2
(
R
)
=
V
j
0
⊕
⊕
j
=
j
0
∞
W
j
¯
.
L
2
(
R
)
=
V
j
0
⊕
⊕
j
=
j
0
∞
W
j
¯
.
(6)
With the notational convention that Wj0-1:=Vj0Wj0-1:=Vj0,
we call the sequence
{Wj}j≥j0-1{Wj}j≥j0-1
a multiscale decomposition (MSD).
We call ψψ a wavelet function whenever the set {ψ(x-k);k∈Z}{ψ(x-k);k∈Z} is a Riesz basis of W0W0. Since W0⊂V1W0⊂V1, there also exist a refinement equation for ψψ, similarly to Equation 3:
ψ
(
x
)
=
2
∑
k
g
k
ϕ
(
2
x
-
k
)
.
ψ
(
x
)
=
2
∑
k
g
k
ϕ
(
2
x
-
k
)
.
(7)
The collection of wavelet functions {ψjk=2j/2ψ(2jx-k);k∈Z,j∈Z}{ψjk=2j/2ψ(2jx-k);k∈Z,j∈Z} is then a Riesz basis for
L2(R)L2(R). One of the main features of the wavelet functions is that they possess a certain number of vanishing moments.
A wavelet function ψ(x)ψ(x) has NNvanishing moments if
∫
ψ
(
x
)
x
p
d
x
=
0
,
p
=
0
,
...
,
N
-
1
.
∫
ψ
(
x
)
x
p
d
x
=
0
,
p
=
0
,
...
,
N
-
1
.
We now mention two interesting cases of wavelet bases.
In an orthogonal multiresolution analysis, the spaces WjWj are defined as the orthogonal complement of VjVj in Vj+1Vj+1.
The following theorem tells us one of the main advantages of such a MRA.
(
[3], Theorem 5.1.1)
If a sequence of closed subspaces
(Vj)j∈Z(Vj)j∈Z in
L2(R)L2(R)
satisfies
Definition 1, and if, in addition,
{ϕ(x-k),k∈Z}{ϕ(x-k),k∈Z} is an orthogonal basis for
V0V0, then there exists one function
ψ(x)ψ(x) such that
{ψ(x-k);k∈Z}{ψ(x-k);k∈Z} forms an orthogonal basis for the orthogonal complement
W0W0 of
V0V0 in
V1V1.
An immediate consequence of Theorem 1 is that
{ψjk,k∈Z}{ψjk,k∈Z} constitutes an orthogonal basis for the orthogonal complement WjWj of VjVj in Vj+1Vj+1.
In this section, let PjPj (resp. QjQj) be the orthogonal projection
operator onto VjVj (resp. WjWj). The orthogonal expansion
f
=
P
j
0
f
+
∑
j
=
j
0
∞
Q
j
f
=
∑
k
f
,
ϕ
j
0
,
k
ϕ
j
0
,
k
+
∑
j
=
j
0
∞
∑
k
f
,
ψ
j
k
ψ
j
k
f
=
P
j
0
f
+
∑
j
=
j
0
∞
Q
j
f
=
∑
k
f
,
ϕ
j
0
,
k
ϕ
j
0
,
k
+
∑
j
=
j
0
∞
∑
k
f
,
ψ
j
k
ψ
j
k
(8)
tells us that a first, coarse approximation of ff in Vj0Vj0 is further refined with the projection of ff onto the detail spaces WjWj.
Figure 1 shows two examples of orthogonal wavelet functions. The first is the Haar wavelet, associated to the Haar scaling function defined in "Definition of subspaces V j and of scaling functions".
ψ
(
x
)
Haar
=
2
-
1
/
2
ϕ
Haar
(
2
x
-
1
)
-
ϕ
Haar
(
2
x
)
=
1
[
1
2
,
1
)
(
x
)
-
1
[
0
,
1
2
)
(
x
)
.
ψ
(
x
)
Haar
=
2
-
1
/
2
ϕ
Haar
(
2
x
-
1
)
-
ϕ
Haar
(
2
x
)
=
1
[
1
2
,
1
)
(
x
)
-
1
[
0
,
1
2
)
(
x
)
.
(9)
The Haar wavelet has only one vanishing moment and consequently is optimal only to represent functions having a low degree of regularity, like, for example, β-β-Hölder functions with 0<β<10<β<1.
Daubechies constructed in [2], [3] compactly supported wavelets which have more than one vanishing moment. Compactly supported wavelets are desirable from a numerical point of view, while having more than one vanishing moment allows to reconstruct exactly polynomials of higher order.
These wavelets cannot, in general, be written in a closed analytic form. However, their graph can be computed with arbitrarily high precision using a subdivision scheme algorithm.
Figure 1(b) represents the Daubechies Least Asymmetric wavelet with N=4N=4 vanishing moments.
This figure also illustrates the reason behind the name `wavelet': since wavelets are functions with a certain number of vanishing moments, they have the shape of a `little wave' or `wavelet'.
Having an orthogonal MRA puts strong constraints on the construction of a wavelet basis.
For example, the Haar wavelet is the only real-valued function which is compactly supported and symmetric.
However, if we relax orthogonality for biorthogonality, then it becomes
possible to have real-valued wavelet bases of fixed but arbitrary high order (see Definition 1 from Approximation of Functions) which are symmetric and compactly supported [1].
In a biorthogonal setting, a dual scaling function ϕ˜ϕ˜ and a dual wavelet function ψ˜ψ˜ exist. They generate a dual MRA with subspaces V˜jV˜j
and complement spaces W˜jW˜j such that
V
˜
j
⊥
W
j
and
V
j
⊥
W
˜
j
.
V
˜
j
⊥
W
j
and
V
j
⊥
W
˜
j
.
(10)
In other words,
ϕ
˜
,
ψ
(
·
-
k
)
=
0
and
ϕ
,
ψ
˜
(
·
-
k
)
=
0
ϕ
˜
,
ψ
(
·
-
k
)
=
0
and
ϕ
,
ψ
˜
(
·
-
k
)
=
0
(11)
Moreover, the dual functions also have to satisfy
ϕ
˜
,
ϕ
(
·
-
k
)
=
δ
k
,
0
and
ψ
˜
,
ψ
(
·
-
k
)
=
δ
k
,
0
,
ϕ
˜
,
ϕ
(
·
-
k
)
=
δ
k
,
0
and
ψ
˜
,
ψ
(
·
-
k
)
=
δ
k
,
0
,
(12)
where δk,0δk,0 is the Kronecker symbol.
By construction, the dual scaling and wavelet functions satisfy a refinement equation, similarly to the equations Equation 3 and Equation 7.
In this work, we use the following convention: the dual MSD will be used to decompose a function (or a signal), while the original, or primal MSD reconstructs the function.
This yields the following representation of a function f∈L2(R)f∈L2(R)
f
(
x
)
=
∑
k
f
,
ϕ
˜
j
0
,
k
ϕ
j
0
,
k
(
x
)
+
∑
j
=
j
0
∞
∑
k
f
,
ψ
˜
j
k
ψ
j
k
(
x
)
.
f
(
x
)
=
∑
k
f
,
ϕ
˜
j
0
,
k
ϕ
j
0
,
k
(
x
)
+
∑
j
=
j
0
∞
∑
k
f
,
ψ
˜
j
k
ψ
j
k
(
x
)
.
(13)
Figure 2 shows an example of a biorthogonal wavelet basis built by Cohen, Daubechies and Feauveau in [1], (called CDF-wavelets hereafter).
-
Cohen, A. and Daubechies, I. and Feauveau., J. (1992). Bi-orthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math., 45, 485-560.
-
Daubechies, I. (1988). Orthonormales bases of compactly supported wavelets. Comm. Pure Appl. Math., 41, 909-996.
-
Daubechies, I. (1992). Ten Lectures on Wavelets. Philadelphia: SIAM.
-
Kovac, A. (1998, March). Wavelet Thresholding for Unequally Spaced Data. Ph. D. Thesis. University of Bristol.
