Graphics Reference
In-Depth Information
h
[
k
] are called the scaling function coecients or the scaling filter
c
oecients.
With the above normalization,
=1and
h
[
k
]=
√
2
<φ
(2
t
h
[
k
]
−
k
)
,φ
(
t
)
>
. Taking the Fourier transform of both sides, we get
φ
(
ω
)=
φ
(
t
)
e
−iωt
dt,
=
√
2
∞
k
)
e
−iωt
,
h
[
k
]
φ
(2
t
−
k
=
−∞
h
[
k
]
φ
(
t
1
)
e
−iω
(
t
1
/
2+
k/
2)
1
∞
=
√
2
2
dt
1
,
(8.32)
k
=
−∞
φ
(
t
1
)
e
−iω/
2
t
1
e
−iω/
2
k
dt
1
,
∞
=
√
2
h
[
k
]
1
2
k
=
−∞
h
[
k
]
e
−i
(
ω/
2)
k
φ
(
t
1
)
e
−i
(
ω/
2)
t
1
∞
1
√
2
=
dt
1
,
k
=
−∞
√
2
h
(
ω/
2)
φ
(
ω/
2)
,
1
=
where
h
(
ω/
2) =
k
h
[
k
]
e
−iω/
2
k
. An important property of
h
(
e
iω
) is the fol-
∈
Z
lowing:
|he
iω
)
|h
(
e
i
(
ω
+
π
)
)
2
+
2
=2
.
|
|
(8.33)
We have already seen that the scaling function
φ
can approximate a func-
tion
f
(
t
) in different subspaces and these subspaces can be obtained by in-
creasing the index
j
, i.e., increasing the size of the subspaces spanned by the
scaling functions. However, this procedure is not ecient and hence, we take
help of wavelet functions at different scales, i.e., at different wavelet subspaces.
The wavelets
ψ
j,r
(
t
) generated from the mother wavelet
ψ
(
t
) span the differ-
ence between the spaces that are spanned by the different scales of the scaling
functions. Scaling functions and wavelets are assumed to be orthogonal for a
number of reasons from the standpoint of computation.
W
j
is defined as the
orthogonal complement of
V
j
in
V
j
+1
, so that all elements of
V
j
are orthogonal
to all elements of
W
j
. For this, we need the following inner product condition
to hold true.
<φ
j,k
(
t
)
,ψ
j,l
(
t
)
>
=
φ
j,k
(
t
)
ψ
j,l
(
t
)
dt,
=0
,
Z.
The wavelet spanned subspace at
j
=0is
V
1
=
V
0
⊕
j,k,l
∈
W
0
. Similarly,
V
2
=
V
1
⊕
W
1
=
V
0
⊕
W
0
⊕
W
1
. Proceeding this way, we finally get
=
L
2
.
···⊕
W
2
⊕
W
1
⊕
W
0
⊕
W
1
⊕
W
2
⊕···
The scaling subspace
V
0
canbeviewedas
W
−∞
⊕···⊕
W
1
=
V
0
.
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