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,
 
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