Graphics Reference
In-Depth Information
8.6 Spline Approximations
Polynomial spline approximations do smooth approximations with fast asym-
totic decay. One can construct a Riesz basis of polynomial splines with box
splines. In this section, we present a slightly different approach to find the
filter coecients. The method is simple and straightforward. A box spline,
α
n
, of order
n
, is computed by convolving the box window 1
[0
,
1]
with itself
n
times. Hence, considering the previous equation
N
n
(
ω
)=(
sin
ω/
2
ω/
2
)
n
e
−iβω/
2
.
When
n
is odd,
β
=1and
α
is centered at
t
=1
/
2, while when
n
is even,
β
= 0, then
α
(
t
) is symmetric about
t
=0.For
n
≥
1,
α
(
t
−
r
)
,r
∈
Z
is a
Riesz basis of
V
0
.
Now, let us consider the following theorem.
Theorem
.Let
Z
be a multiresolution approximation and
φ
be
scaling function whose Fourier transform is
{
V
j
}
,j
∈
N
n
(
ω
)
φ
(
ω
)=
.
(8.35)
∞
| N
n
(
ω
+2
πr
)
2
)
1
/
2
(
|
r
=
−∞
If
φ
j,k
=2
j/
2
φ
(2
j
t
−
k
)
,
then the family
φ
j,k
, ,k
∈
Z
is an orthonormal basis of
V
j
for all
j
∈
Z
.
Proof.
In order to construct an orthonormal basis, we need a function
φ
∈
V
0
that
can be expanded in basis of
N
n
(
t
−
k
), i.e., we must have
φ
(
t
)=
∞
a
[
k
]
N
n
(
t
−
k
)
.
−∞
Taking Fourier transform we get,
φ
(
ω
)=
a
(
ω
)
N
n
(
ω
)
.
a
(
ω
)isa2
π
periodic Fourier series of finite energy. For computation of
a
(
ω
),
. Assuming
φ
(
t
)=
φ
∗
(
we take help of orthogonality of
{
φ
(
t
−
k
)
}
−
t
), we can
write
p
)
>
=
∞
∞
k
)
φ
∗
(
t
<φ
(
t
−
k
)
,φ
(
t
−
φ
(
t
−
−
p
)
dt,
φ
∗
(
p
=
φ
∗
−
k
)
.
φ
(
k
)=
δ
[
k
]. Computing
{
φ
(
t
−
k
)
}
is orthonormal if and only if
φ
∗
Hence,
Fourier transform, we get
Search WWH ::
Custom Search