Graphics Reference
In-Depth Information
d
7
dx
7
cot
x,
S
8
=
−
1
2
8
1
7!
=
−
1
2
8
105
.
48
−
1
16
sin
8
x
[4 cos
6
x
+10cos
4
x
+26
.
4cos
4
x
+26
.
3cos
2
x
+17
.
6cos
2
x
+ 17]
,
=
3sin
2
x
+3sin
4
x
sin
6
x
)
1
2
8
1
105
.
48
16
sin
8
x
[4(1
−
−
+3
.
38 cos
4
x
+3
.
60 cos
2
x
+7]
,
=
105
sin
8
x
[(5 + 30 cos
2
x
+30sin
2
x
cos
x
)
+ (70 cos
4
x
+2sin
4
x
cos
2
x
+
3
sin
6
x
)]
.
1
2
8
1
8.6.1 Battle-Lemarie Wavelets
Battle and Lemarie wavelets are polynomial splines. These wavelets can be
computed from multiresolution approximation. To get these wavelets in a
general form, one can consider splines of order
n
for which
h
(
ω
) and first
n
1 derivatives are zero at
ω
=
π
. The wavelet
ψ
has
n
vanishing moments.
Being a polynomial of order
n
,ithasdegree
n
−
−
1 and hence it is
n
−
2 times
continuously differentiable. Also, when the degree of the polynomial is odd,
ψ
is symmetric about
1
2
and when the degree is even,
ψ
is antisymmetric about
1
2
.
From equation (8.34),
ψ
(
ω
)=
ψ
(
t
)
e
−iωt
dt,
=
√
2
∞
k
)
e
−iωt
,
h
1
[
k
]
φ
(2
t
−
k
=
−∞
h
1
[
k
]
φ
(
t
1
)
e
−iω
(
t
1
/
2+
k/
2)
1
∞
=
√
2
2
dt
1
,
(8.39)
k
=
−∞
φ
(
t
1
)
e
−iω/
2
t
1
e
−iω/
2
k
dt
1
,
∞
=
√
2
h
1
[
k
]
1
2
k
=
−∞
h
1
[
k
]
e
−i
(
ω/
2)
k
φ
(
t
1
)
e
−i
(
ω/
2)
t
1
∞
1
√
2
=
dt
1
,
k
=
−∞
√
2
h
1
(
ω/
2)
φ
(
ω/
2)
,
1
=
where,
h
1
(
ω
)=
k
h
[
k
]
e
−iωk
, and for any scale 2
j
,
{
ψ
j,r
}
,j,r
∈
Z
is an
∈
Z
orthonormal basis of
L
2
(
IR
).
h
1
(
ω
) is connected to
h
(
ω
) through
h
1
(
ω
)=
e
−iω
h
∗
(
ω
+
π
)
.
Mallat and Meyer [116] proved that
{
ψ
j,r
}
,r
∈
Z
is an orthonormal basis of
W
j
if and only if
|h
1
(
ω
)
|h
1
(
ω
+
π
)
2
+
2
=2
|
|
and
Search WWH ::
Custom Search