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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
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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