Image Processing Reference
In-Depth Information
4.2 Estimation of Fourier coefficients in experimental design
First, we present the following theorem (Ukita et al., 2010a).
n
Domain
Theorem 1.
Sampling Theorem for Bandlimited Functions over a GF
(
q
)
n
is monotonic and
Assume that A
⊆{
0, 1
}
(
x
)=
a
∈
I
A
f
a
X
a
(
x
)
,
f
(27)
where I
A
=
{
(
b
1
a
1
,...,
b
n
a
n
)
|
a
∈
A
,
b
i
∈
GF
(
q
)
}
. Then, the Fourier coefficients can be computed
as follows:
1
q
k
∑
x
∈
C
⊥
(
x
)
X
a
(
x
)
a
=
f
f
,
(28)
where C
⊥
is an orthogonal design for A (
C
⊥
|
=
q
k
).
|
When an experiment is conducted in accordance to the orthogonal design
C
⊥
, unbiased
estimators of
f
(
x
)=
in (26) can be obtained by using Theorem 1 and assuming that
E
0:
a
1
q
k
f
∑
x
∈
C
⊥
(
x
)
X
a
(
x
)
a
=
y
.
(29)
Then, the Fourier coefficients can be easily estimated by using Fourier transform. There are
a number of software packages for Fourier transform, which can be used to calculate (29) for
any monotonic set
A
.
Example 10.
Consider the case that a set A is given by (14) and the result of experiments is given by
Table 1. Then,
X
a
(
x
)=
e
−
2
π
i
(
a
1
x
1
+
a
2
x
2
+
a
3
x
3
+
a
4
x
4
+
a
5
x
5
)
/3
.
(30)
Using (29), (30) and e
2
π
ik
=
1
for any integer k,
f
00000
=
2596/27,
f
10000
=(
954
e
−
2
π
i
/3
779
e
−
4
π
i
/3
863
+
+
)
/27,
f
20000
779
e
−
2
π
i
/3
954
e
−
4
π
i
/3
=(
+
+
)
863
/27,
f
01000
873
e
−
2
π
i
/3
852
e
−
4
π
i
/3
=(
871
+
+
)
/27,
f
02000
=(
852
e
−
2
π
i
/3
873
e
−
4
π
i
/3
871
+
+
)
/27,
f
00100
868
e
−
2
π
i
/3
886
e
−
4
π
i
/3
=(
+
+
)
842
/27,
f
00200
886
e
−
2
π
i
/3
868
e
−
4
π
i
/3
=(
842
+
+
)
/27,
f
00010
=(
848
e
−
2
π
i
/3
875
e
−
4
π
i
/3
873
+
+
)
/27,
f
00020
875
e
−
2
π
i
/3
848
e
−
4
π
i
/3
=(
+
+
)
873
/27,
f
00001
=(
864
e
−
2
π
i
/3
885
e
−
4
π
i
/3
847
+
+
)
/27,
f
00002
885
e
−
2
π
i
/3
864
e
−
4
π
i
/3
=(
+
+
)
847
/27,
f
11000
863
e
−
2
π
i
/3
874
e
−
4
π
i
/3
=(
859
+
+
)
/27,
f
12000
=(
868
e
−
2
π
i
/3
861
e
−
4
π
i
/3
867
+
+
)
/27,
f
21000
861
e
−
2
π
i
/3
868
e
−
4
π
i
/3
=(
+
+
)
867
/27,
f
22000
=(
874
e
−
2
π
i
/3
863
e
−
4
π
i
/3
859
+
+
)
/27,
f
10100
=(
861
e
−
2
π
i
/3
875
e
−
4
π
i
/3
860
+
+
)
/27,
f
10200
857
e
−
2
π
i
/3
868
e
−
4
π
i
/3
=(
+
+
)
871
/27,