Image Processing Reference
In-Depth Information
3.2 Orthogonal design
Definition 2. (Orthogonal design)
Define v
n , let H A be the k
( a )= {
i
|
a i
=
0, 1
i
n
}
. For A
⊆{
0, 1
}
×
n matrix
h 11 h 12 ... h 1 n
h 21 h 22 ... h 2 n
. . . . . .
h k 1 h k 2 ... h kn
H A =
.
(12)
The components of this matrix, h ij
GF
(
q
)(
1
i
k ,1
j
n
)
, satisfy the following conditions.
( a + a ) }
1 , where
{ h · j |
1. The set
j
v
h · j is the j-th column of H A , is linearly independent over
a ,
a
GF
(
q
)
for any given
A.
{ h i · |
}
(
)
2. The set
1
i
k
, where
h i ·
is the i-th row of H A , is linearly independent over GF
q
.
An orthogonal design C for main and interactive factors A
n is defined as
⊆{
}
0, 1
C = { x | x = r
k
r
(
)
}
H A ,
GF
q
,
(13)
C | =
q k .
and
|
Example 6. We consider the case q
=
3, n
=
5 and
A
= {
00000, 10000, 01000, 00100, 00010, 00001, 11000, 10100, 10010
}
.
(14)
In this case,
,
10000
01011
00112
H A =
(15)
satisfies the conditions in Definition 2. Therefore,
C = {
00000, 00112, 00221, 01011, 01120, 01202, 02022, 02101, 02210, 10000, 10112,
10221, 11011, 11120, 11202, 12022, 12101, 12210, 20000, 20111, 20221, 21011,
21120, 21202, 22022, 22101, 22210
}
,
is an orthogonal design for A.
Many algorithms for constructing H A have been proposed (Hedayat et al., 1999; MacWilliams
& Sloane, 1977; Takahashi, 1979; Ukita et al., 2003). However, it is still an extremely difficult
problem to construct H A when the number of factors n is large and a large number of
interactions are included in the model. In this regard, algorithms for the construction of
orthogonal design are not presented here since this falls outside the scope of this chapter.
1 For
n , the addition of vectors
a 1
=(
a 11 , a 12 ,..., a 1 n
)
,
a 2
=(
a 21 , a 22 ,..., a 2 n
) ∈{
0, 1
}
a 1 and
a 2 is defined
+ a 2
=(
)
as
a 1
a 11
a 21 , a 12
a 22 ,..., a 1 n
a 2 n
, where
is the exclusive OR operator.
 
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