Image Processing Reference
In-Depth Information
Example 12. Consider the case that a set A is given by (14) and the result of experiments is given by
Table 1. Then, using the result of Example10,
k
0 e 2 π ik /3
0 and e 2 π i
=
=
1 ,
=
f 00000
2
f 00000
f 00000 =(
|
|
=
)(
)=
2596/27
2596/27
9244.466,
(42)
f 10000 |
2
f 10000
f 10000
|
=
954 e 2 π i /3
779 e 4 π i /3
954 e 2 π i /3
779 e 4 π i /3
/27 2
=(
863
+
+
)(
863
+
+
)
863 2
954 2
779 2
/27 2
=(
+
+
863
·
954
863
·
779
954
·
779
)
=
31.52401.
(43)
Similarly,
f 20000 |
2
|
=
31.52401,
f 01000
f 02000
f 00100
f 00200
2
2
2
2
|
|
= |
|
=
|
|
= |
|
=
0.552812,
2.013717,
f 00010
f 00020
f 00001
f 00002
2
2
2
2
|
|
= |
|
=
0.931413,
|
|
= |
|
=
1.491084,
f 11000 |
f 22000 |
f 12000 |
f 21000 |
2
2
2
2
|
= |
=
0.248285,
|
= |
=
0.058985,
f 10100
f 20200
f 10200
f 20100
2
2
2
2
|
|
= |
|
=
|
|
= |
|
=
0.289438,
0.223594,
f 10010 |
f 20020 |
f 10020 |
f 20010 |
2
2
2
2
|
= |
=
0.548697,
|
= |
=
0.577503.
Hence, using Theorem 5-7 and (41),
f 00000
2
=
|
|
=
SS Mean
27
249600.6,
f 10000 |
f 20000 |
2
2
SS F 1 =
27
( |
+ |
)=
1702.3,
f 01000 |
2
f 02000
2
SS F 2 =
( |
+ |
|
)=
27
29.9,
f 00100
f 00200
2
2
SS F 3 =
27
( |
|
+ |
|
)=
108.7,
f 00010 |
f 00020 |
2
2
SS F 4 =
27
( |
+ |
)=
50.3,
f 00001
f 00002
2
2
SS F 5 =
( |
|
+ |
|
)=
27
80.5,
f 11000 |
f 12000 |
f 21000 |
f 22000 |
2
2
2
2
SS F 1 × F 2 =
27
( |
+ |
+ |
+ |
)=
16.6,
f 10100 |
f 10200 |
f 20100 |
f 20200 |
2
2
2
2
SS F 1 × F 3 =
27
( |
+ |
+ |
+ |
)=
27.7,
f 10010
f 10020
f 20010
f 20020
2
2
2
2
SS F 1 × F 4 =
( |
|
+ |
|
+ |
|
+ |
|
)=
27
60.8,
=
SS Error
16.6.
Therefore,
the
analysis
of
variance
can
be
executed
in
the
proposed
description
of
experimental design.
Hence, it is clear that two main procedures in the experimental design, that is, the estimation
of the effects and the analysis of variance can be executed in a description of experimental
design on the basis of an orthonormal system.
5. Conclusion
In this chapter, we have proposed that the model of experimental design be expressed as an
orthonormal system, and shown that the model contains no redundant parameters. Then, the
model is expressed by using Fourier coefficients instead of the effect of each factor. As there
is an abundance of software for calculating the Fourier transform, such a system allows for a
straightforward implementation of the procedures for estimating the Fourier coefficients by
using Fourier transform. In addition, the effect of each factor can be easily obtained from the
Fourier coefficients. Therefore, it is possible to implement easily the estimation procedures
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