Image Processing Reference
In-Depth Information
q
1
ϕ = 0
q
1
ψ = 0 Y l , m ( ϕ , ψ ) SS F l SS F m SS Mean ,
1
q k 2
SS F l × F m =
(24)
SS Error =
x C
y 2
SS Mean
l
( x )
SS F l
SS F l × F m .
(25)
MF
{
l , m
}∈
IF
Example 8. Consider the case that a set A is given by (14) and the result of experiments is given by
Table 1. Then, using (22)-(25),
SS Mean
=
249600.6, SS F 1
=
1702.3, SS F 2
=
29.9, SS F 3
=
108.7,
SS F 4
=
50.3, SS F 5
=
80.5, SS F 1 × F 2 =
16.6, SS F 1 × F 3 =
27.7,
SS F 1 × F 4 =
=
60.8, SS Error
16.6.
4. Description of experimental design on the basis of an orthonormal system
In this section, we propose the model of experimental design on the basis of an orthonormal
system.
4.1 Model on the basis of an orthonormal system in experimental design
We use y
( x )
to denote the response of an experiment with a level combination
x
, and assume
the following model:
( x )= a I A f a X a ( x )+ x ,
y
(26)
= { (
) | a
(
) }
where I A
b 1 a 1 ,..., b n a n
A , b i
GF
q
and
x
is a random error with a zero mean
and a constant variance.
Then, the model is expressed by using Fourier coefficients instead of the effect of each
factor.
{
a | a
}
. In addition, there are
no constraints between the parameters, and the parameters are independent. Hence, it is clear
that the model contains no redundant parameters.
The effects are represented by the parameters
f
I A
Example 9. Consider q
=
3, n
=
5 and A
= {
00000, 10000, 01000, 00100, 00010, 00001, 11000,
10100, 10010
}
. Then, I A is given by
I A = {
00000, 10000, 20000, 01000, 02000, 00100, 00200, 00010, 00020, 00001, 00002, 11000,
12000, 21000, 22000, 10100, 10200, 20100, 20200, 10010, 10020, 20010, 20020
}
,
and Fourier coefficients
f 00000 , f 10000 , f 20000 , f 01000 , f 02000 , f 00100 , f 00200 , f 00010 , f 00020 , f 00001 , f 00002 , f 11000 , f 12000 , f 21000 ,
f 22000 , f 10100 , f 10200 , f 20100 , f 20200 , f 10010 , f 10020 , f 20010 , f 20020
are parameters. The number of parameters is 23 , and these parameters are independent.
 
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