Image Processing Reference
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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.
This chapter is organized as follows. In Section 2, we give preliminaries that are necessary
for this study. In Section 3, we provide an introduction to experimental design and describe
the characteristic of the previous model in .experimental design. In Section 4, we propose
the new model of experimental design on the basis of an orthonormal system and clarify the
characteristic of the model. Finally, Section 5 concludes this chapter.
2. Preliminaries
2.1 Fourier analysis on finite Abelian groups
Here, we provide a brief explanation of Fourier analysis on finite Abelian groups. Characters
are important in the context of finite Fourier series.
2.1.1 Characters
Let G be a finite Abelian group (with additive notation), and let S 1
be the unit circle in the
S 1
complex plane. A character on G is a complex-valued function
X
: G
that satisfies the
condition
X ( x + x )= X ( x ) X ( x ) x
x
,
G .
(1)
In other words, a character is a homomorphism from G to the circle group.
2.1.2 Fourier transform
Let G i , i
=
1, 2, . . . , n , be Abelian groups of respective orders
|
G i | =
g i , i
=
1, 2, . . . , n , g 1
g 2 ≤···≤
g n , and let
n
i = 1 g i .
n
i
= ×
=
G
1 G i
and g
(2)
=
Since the character group of G is isomorphic to G , we can index the characters by the elements
of G , that is,
{X a ( x ) | a
}
X 0 ( x )
G
are the characters of G . Note that
is the principal character
{X a ( x ) | a
}
and identically equal to 1. The characters
G
form an orthonormal system:
1,
1
g
a = b
,
x G X a ( x ) X b ( x )=
(3)
0,
a = b
,
X
where
b ( x )
is the complex conjugate of
X b ( x )
.
C
Any function f : G
is the field of complex numbers, can be uniquely expressed
as a linear combination of the following characters:
, where
C
( x )= a G f a X a ( x ) ,
f
(4)
where the complex number
1
g
x G f ( x ) X a ( x )
f
a =
(5)
is the
a
-th Fourier coefficient of f .
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