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2 The Two-Dimensional Multiresolution Analysis
Firstly, we introduce multiresolution analysis of space
LR
Wavelets can be con-
structed by means of multiresolution analysis. In particular, the existence theorem
[8]
for higher-dimentional wavelets with arbitrary dilation matrice has been given. Let
2
2
(
.
2
satisfy the following refinement equation:
hx LR
()
∈
(
2
)
∑
hx a
()
=⋅
2
b hax n
⋅
(
−
)
(2)
n
2
nZ
∈
where
hx
is
called scaling function. Formula (1) is said to be two-scale refinement equation. The
frequency form of formula (1) can be written as
{()}
nZ
bn
∈
is real number sequence which has only finite terms.and ( )
2
12
h Bz z h
()
ω
=
(, )(
ω
a
,
(3)
where
∑
b
⋅
zz
n
⋅
n
Bz z
(, )
nn Z
=
(, )
nn
.
(4)
1
2
1
2
12
1
2
2
(,
)
∈
12
2
2
j
XLRj Z
⊂
()(
∈
)
Define a subspace
by
j
j
2
X ls ahannZ
=
(
⋅ −
) :
∈
.
(5)
j
22
LR
()
hx
in (2) generate a multiresolution analysis
{}
j
()
Definition 2.
We say that
of
X
j Z
∈
LR
, if the sequence
{}
j
2
()
2
X
defined in (4) satisfy the following properties:
j Z
∈
XX j Z
⊂
1
,
∀
∈
;
∩
∪
is dense in
(i)
(ii)
X
=
{0};
X
LR
; (iii)
2
()
j
j
+
j
j
jZ
∈
jZ
∈
fx X fax X k Z
+
()
∈⇔
( )
∈
,
∀∈
;
(iv) the family
{(
j
hax n n Z
−
):
∈
2
}
forms a
k
k
1
X
.
Riesz
basis for the spaces
k
Yk Z
(
∈
)
X
in
X
+
Let
denote the complementary subspace of
, and assume
1
that there exist a vector-valued function
Ψ=
() { (),
x
ψ
x
ψψ
(),
x
()}
x
constitutes a
1
2
3
Y
, i.e.,
Riesz
basis for
W ls
=
λ
ψλ
:
=
, ,
,
a nZ
2
−
;
∈
2
,
(6)
j
22
:,
j n
LR
()
where
jZ
∈
, and
ψ
:,
()
j j
jk
xa axn
=
/2
ψ
(
−
),
λ =
1, 2 ,
,
a
2
−
1;
n Z
∈
2
.
Form con-
λ
λ
ψ
(),
x
ψ
(),
x
ψ
3
()
x
YX
⊂
1
.
dition (5), it is obvious that
are in
Hence there ex-
1
2
0
ist three real number sequences
{ (
q
()
λ
λ ∈Δ =
, ,
,
a
2
−
,
nZ
∈
2
)
such that
n
∑
2
(
λ
)
ψ
()
xa qh xn
=⋅
(
−
),
(7)
λ
n
2
nZ
∈
Formula (7) in frequency domain can be written as
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