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
.