Biomedical Engineering Reference
In-Depth Information
Let
{
V
m
}
be a multiresolution of
L
2
(
R
); a separable two-dimensional mul-
tiresolution is composed of the tensor product spaces
V
m
=
V
m
⊗
V
m
.
The space
V
m
is the set of the finite energy functions that are linear expan-
sions of the set of the separable bases
φ
m
,
k
,
l
(
x
,
y
)
k
,
l
=
0
,
while the correspondent
wavelet subspace
W
m
is given by the close linear span of
φ
m
,
k
(
x
)
ψ
m
,
l
(
y
)
,ψ
m
,
l
(
x
)
φ
m
,
k
(
y
)
,ψ
m
,
k
(
x
)
ψ
m
,
l
(
y
)
k
,
l
=
0
where
φ
m
,
k
(
x
):
=
2
2
φ
(2
m
x
−
k
)
,
(5.68)
m
2
ψ
(2
m
x
−
k
)
,
φ
m
,
k
,
l
(
x
,
y
):
=
φ
m
,
k
(
x
)
φ
m
,
l
(
y
)
.
ψ
m
,
k
(
x
):
=
2
Like in 1D case, we have
V
m
=
V
m
−
1
⊕
W
m
−
1
=
(
V
m
⊗
V
m
)
⊕
W
m
−
1
,
W
m
=
(
V
m
⊗
W
m
)
⊕
(
W
m
⊗
V
m
)
⊕
(
W
m
⊗
W
m
)
,
and
L
2
(
R
2
)
=⊕
m
=−∞
W
m
.
Wells
et al.
[67] proved the following theorem.
Theorem 8 (Wells and Zhou).
Assume the function f
∈
C
2
(
)
,
where
is
a bounded open set in R
2
.
Let
2
m
k
,
l
∈
1
f
(
k
+
c
2
j
l
+
c
2
j
f
m
(
x
,
y
):
=
)
φ
m
,
k
(
x
)
φ
m
,
l
(
y
)
,
x
,
y
∈
,
(5.69)
,
where
=
k
∈
Z
:
supp
(
φ
m
,
k
)
∩
=∅
is the index set and
2
N
−
1
1
√
2
c
=
kc
k
.
k
=
0
Then
||
f
−
f
m
||
L
2
(
)
≤
C
(1
/
2
m
)
2
,
where C is dependent on the diameter of
,
the first and second moduli of the
first- and second-order derivatives of f on
.
Formula (5.69) is the one which was used in the wavelet-based SFS method.
Now we are ready to introduce this method.
