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ψω
ˆ ()
=
Qzzha
()
λ
(, )(
ω
,
λ
=
1,2, ,
a
2
−
1.
(8)
12
()
λ
2
2
where the signal of sequence
{ (
q
λ =
, ,
,
a
−
,
k Z
∈
)
is
∑
n
n
Qzz q zz
()
λ
(, )
=
()
λ
⋅
⋅
.
(9)
1
2
12
(
nn
,
)
1
2
12
2
(,
nn Z
)
∈
12
ht
()
∈
A bivariate function
2 2
L(R )
is called a semiorthogonal one, if
hhn
δ
(), (
⋅
⋅ − =
)
nZ
∈
2
.
,
(10)
0,
n
(
()
T
)
We say
is a semiorthogonal bivariate vector-valued
wavelets associated with the scaling function
Ψ=
()
x
ψψ ψ
(),
x
(),
x
x
1
2
3
ht
, if they satisfy:
()
h
(),
⋅
ψ
(
⋅ −=
n
)
0
ν ∈Δ
,
nZ
∈
2
,
, (11)
ν
ψ
(),
⋅
ψ
(
⋅ −=
n
)
δ
δ
,
λν∈Δ
,
,
nZ
∈
2
(12)
λ
ν
λ ν
,
,
n
3 The Traits of Nonseparable Bivariate Wavelet Packs
To construct wavelet packs, we introduce the following notation:
a
=Λ
2,
( )
x
=
h x
( ),
0
()
()
ν ∈Δ
0
ν
We are now in a posi-
tion of introdu- cing orthogonal trivariate nonseparable wavelet packets.
Λ=
()
x
ψ
(),
x b
()
k
=
b k
(),
bkqk
()
=
()
ν
(),
where
ν
ν
3,
⋅⋅⋅
,
ν ∈Δ
}
Definition 3.
A family of functions
is called a
nonseparable bivariate wavelet packs with respect to the semiorthogonal scaling
function
{
Λ
n
xn
( ) :
=
0,1, 2,
4
+
Λ
0
()
x
, where
∑
()
ν
Λ
()
x
=
b
()
k
Λ
(2
x
−
k
),
(13)
4
n
+
2
n
kZ
∈
ν
=
0,1, 2, 3.
where
By taaking the Fourier transform for the both sides of (12), we
have
()
(
)
()
ν
Λ
ω
=
Bzz
(, )
⋅ Λ
ω
2.
(14)
4
n
+
ν
n
12
where
(
)
(
)
=
∑
()
ν
()
ν
()
ν
( )
kk
BzzB
,
=
ω
/ 2
bkzz
(15)
1
2
12
12
2
kZ
∈
()
x
x
is an orthogonal one if and only if
()
Lemma 1
[6]
. Let
∈
L(R ).
2
2
Then
∑
ˆ
(
)
2
|
ωπ
+
2
k
|
=1
. (16)
2
kZ
∈
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