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wavelet packs attributes to Cohen and Daubechies.Tensor product multivariate wa-
velet packs has been constructed by Coifman and Meyer. The introduction for the no-
tion on nontensor product wavelet packs attributes to Shen [7]. Since the majority of
information is multidimensional information, many researchers interest themselves in
the investigation into multivariate wavelet theory. But, there exist a lot of obvious de-
fects in this method, such as, scarcity of designing freedom. Therefore, it is signifi-cant
to investigate nonseparable multivariate wavelet theory. Nowadays, since there is little
literature on biorthogonal wavelet packs, it is necessary to investigate biorthogo-nal
wavelet packs. The notion for nonseparable orthogonal quarternary wavelet packs is
given and a procedure for constructing them is described. Next, the biorthogonality
property of nonseparable quarternary wavelet packs is studied.Wavelet packs, owing to
their nice characteristics, have been widely applied to signal Yang [8] constructed a-
scale orthogonal multiwavelet wraps that were more flexible in applications. It is
known that the majority of information is multi-dimensional information. Shen intro-
duced multivariate orthogonal wavelets which may be used in a wider field. Thus, it is
necessary to generalize the concept of multivariate wavelet wraps to the case of quarte-
rnary nonseparable vector-valued wavelets. The goal of this paper is to give the definit-
ion and the constructing procedure of orthogonal quarternary wavelet wraps and
characterize their properties.
2 The Four-Dimensional General Multiresolution Analysis
We start from the following notations.
Z
and
N
stand for integers and nonnegative
integers, respectively. Let
R
be the set of all real numbers.
4
R
denotes the 4- dimensi-
L(R )
,
we denote the square integrable function space on
2
4
onal
Euclidean
space. By
xxxxx
=
(, , , )
uuuuu
=
(, , , ,
ωωωω
=
( , , ,
ω
4
),
z
ι
R . Set
4
∈
R
,
1234
1234
1 2 3
−
i
ω
2
e
ι =
ϕ
,
∈
2 4
L(R ) and the
=
, where
1, 2, 3, 4
.The inner product for any
() ()
χ
χ
ϕ
()
Fourier transform of
χ
are defined as, respectively
,
∫
∫
−⋅
i
χω
ϕ
,
=
ϕχ
(
)
(
χ
)
d
χ
,
(
ω
)
=
(
χ
)
e
d
χ
4
4
R
R
where
ωωωωω
⋅=
x
x
+
x
+
x
+
x
= and ()
denotes the conjugate. There exist
χ
11
22
33
44
4
Z
+
256
elements
μ
,
μμ
,
,
in
by finite group theory such that
0
1
255
4
4
Γ=
{,
ν
a
=
det(
A
),
4
4
)
;
(
ν
+
AZ
)
∩
(
ν
+
AZ
)
=
Ø
, where
Z
=
∪
(
μ
+
AZ
1
2
0
μ
∈Γ
0
ν
1
,
,
255
}
ν
denotes the set of all different representative elements in the quotient
4
4
Z
/(
AZ
)
ν
,
ν
Γ
0
,
A
is a
group
and
denote two arbitrary distinct elements in
12
4
Z
+
44
×
ν
=
0
, where
0
is the null of
Γ=Γ −
{
0
}
integer matrix. Set
0
. Let
0
v
LRC
, we denote the aggregate of all vector-
2
4
(,
ΓΓ
,
)
and
be two index sets. By
0
v
LRC
2
4
T
valued functions
Φ
( ),
x
that is,
(,
):
=Φ
{()(( ,
x
=∅
x
∅
2
(), ,
x
∅
()) :
x
1
v
v
, where
T
means the transpose of a vector. Video
images and digital films are examples of vector-valued functions where
2
4
∅∈
l
xLRl
()
( ),
=
1,2, ,
∅
l
x
()
in the
denotes the pixel on the
l
th column at the point
x
.
above ( )
Φ
x
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