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 L2(IR)L2(IR) is defined as a sequence of closed subspaces Vj⊂L2(IR),j∈ZZVj⊂L2(IR),j∈ZZ with the following properties:
-
...
⊂
V
-
1
⊂
V
0
⊂
V
1
...
...
⊂
V
-
1
⊂
V
0
⊂
V
1
...
(1)
- The spaces VjVj satisfy
⋃j∈ZZVjisdenseinL2(IR)and⋂j∈ZZVj={0}⋃j∈ZZVjisdenseinL2(IR)and⋂j∈ZZVj={0}(2)
- 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.
- If f∈V0,f(.-k)∈V0,k∈ZZ.f∈V0,f(.-k)∈V0,k∈ZZ.
That is, V0V0(and hence all the VjVj) is invariant under translation.
- There exists ϕ∈V0ϕ∈V0 such that {ϕ0,n;n∈ZZ}{ϕ0,n;n∈ZZ} 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).
Let us make a couple of simple observations concerning this definition. Combining the facts that
-
ϕ
(
x
)
∈
V
0
ϕ
(
x
)
∈
V
0
(3)
- {ϕ(.-k),k∈ZZ}{ϕ(.-k),k∈ZZ} is an orthonormal basis for V0V0
- ϕ(2jx)∈Vjϕ(2jx)∈Vj,
we obtain that, for fixed jj,
{ϕj,k(x)=2j/2ϕ(2jx-k),k∈ZZ}{ϕj,k(x)=2j/2ϕ(2jx-k),k∈ZZ} is an orthonormal basis for VjVj.
Since ϕ∈V0⊂V1,ϕ∈V0⊂V1, 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>.
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)
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,k∈ZZ}{ψj,k,k∈ZZ} 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.
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 j∈ZZj∈ZZ 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 l∈ZZ,l∈ZZ,
since the other conditions simply follow from scaling.
An orthogonal wavelet is a function ψψ such that the collection of
functions
{ψ(x-l)|l∈ZZ}{ψ(x-l)|l∈ZZ} 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.
Suppose we have an orthonormal basis (ONB) {ϕj,k,k∈ZZ}{ϕj,k,k∈ZZ} for VjVj and we want to construct ψjkψjk such that
- ψjk,k∈ZZψjk,k∈ZZ 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:
- Since ϕ∈V0⊂V1,ϕ∈V0⊂V1, and the ϕ1,kϕ1,k are an ONB in V1,V1, we have:
ϕ(x)=∑khkϕ1,k,hk=<ϕ,ϕ1,k>,∑k∈ZZ|hk|2=1.ϕ(x)=∑khkϕ1,k,hk=<ϕ,ϕ1,k>,∑k∈ZZ|hk|2=1.(21)
(refinement equation)
δk,0=∫ϕ(x)ϕ(x-k)¯dxδk,0=∫ϕ(x)ϕ(x-k)¯dx(22)
(orthonormality of ϕ(.-k)ϕ(.-k))
- Let us now characterize W0:f∈W0W0:f∈W0 is equivalent to f∈V1f∈V1 and f⊥V0.f⊥V0. Since f∈V1,f∈V1, we have:
f=∑nfnϕ1,n,withfn=<f,ϕ1,n>.f=∑nfnϕ1,n,withfn=<f,ϕ1,n>.(23)
The constraint f⊥V0f⊥V0 is implied by f⊥ϕ0,kf⊥ϕ0,k for all k.k. - Taking the general form of f∈W0,f∈W0, 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 Entry 1, chapter 5). Let us now state the result of these
manipulations.
(Daubechies, chap 5)
If a ladder of closed subspaces
(Vj)j∈ZZ(Vj)j∈ZZ in
L2(IR)L2(IR) satisfies the conditions of the
Definition 1, then there exists an associated orthonormal wavelet basis
{ψj,k;j,k∈ZZ}{ψj,k;j,k∈ZZ} 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).
Let us see what the recipe of Theorem 1 gives for the Haar multiresolution analysis. In that case, ϕ(x)=1ϕ(x)=1 for 0≤x≤1,00≤x≤1,0 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
