Image Processing Reference
In-Depth Information
n
Assume that q is a prime power. Let GF
2.2 Fourier analysis on GF
(
q
)
(
q
)
be a Galois field of order q which contains a finite
n to denote the set of all n -tuples with entries from
number of elements. We also use GF
(
q
)
n are referred to as vectors.
GF
(
q
)
. The elements of GF
(
q
)
Example 1. Consider GF
(
3
)= {
0, 1, 2
}
. Addition and multiplication are defined as follows:
+
·
012
0 012
1 120
2 201
012
0 000
1 012
2 021
Moreover, consider n
=
5 .
5
(
)
= {
···
}
GF
3
00000, 10000,
, 22222
,
(6)
5
and
|
GF
(
3
)
| =
243 .
n and g
q n , the
Specifying the group G in Section 2.1.2 to be the support group of GF
(
q
)
=
n domain.
relations (3), (4) and (5) also hold over the GF
(
q
)
3. Experimental design
In this section, we provide a short introduction to experimental design.
3.1 Model in experimental design
Let F 1 , F 2 ,..., F n denote n factors to be included in an experiment. The levels of each factor can
be represented by GF
(
q
)
, and the combinations of levels can be represented by the n -tuples
n .
x =(
x 1 , x 2 ,..., x n
)
GF
(
q
)
Example 2. Let Machine (F 1 ) and Worker (F 2 ) be factors that might influence the total amount of the
product. Assume each factor has two levels.
F 1 : new machine (level 0 ), old machine (level 1 ).
F 2 : skilled worker (level 0 ), unskilled worker (level 1 ).
For example,
01 represents a combination of new machine and unskilled worker.
Then, the effect of the machine, averaged over both workers, is referred to as the effect of main factor F 1 .
Similarly, the effect of the worker, averaged over both machines, is referred to as the effect of main factor
F 2 . The difference between the effect of the machine for an unskilled worker and that for a skilled worker
is referred to as the effect of the interaction of F 1 and F 2 .
x =
n represent all factors that might influence the response of an experiment.
The Hamming weight w
Let the set A
⊆{
0, 1
}
( a )
a =(
)
A is defined as the number of
nonzero components. The main factors are represented by MF
of a vector
a 1 , a 2 ,..., a n
= {
|
a l =
a
A 1 }
l
1,
, where
= { a |
( a )=
a
}
= {{
}|
a l =
A 1
w
1,
A
. The interactive factors are represented by IF
l , m
=
a
}
= { a |
( a )=
a
}
1, a m
1,
A 2
, where A 2
w
2,
A
.
= {
}
= {
}
=
Example 3. Consider A
000, 100, 010, 001, 110
. Then, A 1
100, 010, 001
and MF
{
}
= {
}
= {{
}}
1, 2, 3
,A 2
110
and IF
1, 2
.
{
}∈
For example, 1
MF indicates the main factor F 1 , and
1, 2
IF indicates the interactive factors F 1
and F 2 .
 
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