Graphics Reference
In-Depth Information
8.5 Multiresolution Analysis and Wavelet Bases
The concept of multiresolution analysis was first published in 1989 by Mallat
[115] and Meyer in 1990 [125]. Here the main objective is to find a function
ψ
such that
{
ψ
j,r
}
is an orthonormal basis of
L
2
(
IR
). In
{
ψ
j,r
}
, we have two
parameters: one is the translation parameter and the other is the dilation
parameter designated respectively by
r
and
j
. Now, considering the Fourier
transform, we can write
| ψ
(
ω
| ψ
j,r
(
ω
)
=2
−j/
2
|
2
j
)
|
.
Therefore, for fixed
j
, we get a fixed bandwidth in the signal.
Definition (MRA)
: A multiresolution analysis consists of a sequence of
embedded closed subspaces
···
V
2
⊂
V
1
⊂
V
0
⊂
V
1
⊂
V
2
···
(8.25)
such that we have
(1) Upward completeness:
j∈Z
V
j
=
L
2
(
IR
)
(8.26)
(2) Downward completeness:
j∈Z
V
j
=
{
}
0
(8.27)
(3) Scale invariance:
f
(2
j
f
(
t
)
∈
V
j
⇐⇒
t
)
∈
V
j
+1
(8.28)
(4) Shift invariance:
f
(
t
)
∈
V
0
=
⇒
f
(
t
−
r
)
∈
V
0
∀
r
∈
Z
(8.29)
(5) Existence of a basis: There exists
φ
∈
V
0
, such that
{
φ
(
t
−
r
)
|
r
∈
Z
}
(8.30)
is an orthonaormal basis for
V
0
. Because of the embedding spaces of functions
(equation(8.25)) and the scaling property (equation(8.28)), one can verify that
the scaling function
φ
(
t
) satisfies a two-scale equation. Since
V
0
is included
in
V
1
,
φ
(
t
), which belongs to
V
0
, belongs to
V
1
as well. As such,
φ
(
t
)canbe
written as a linear combination of the weighted sum of shifted
φ
(2
t
). Thus
φ
(
t
) can be expressed as
∞
φ
(
t
)=
√
2
h
[
k
]
φ
(2
t
−
k
)
k
∈
Z.
(8.31)
k
=
−∞
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