Image Processing Reference
In-Depth Information
f 20100
868 e 2 π i /3
857 e 4 π i /3
=(
+
+
)
871
/27,
f 20200
875 e 2 π i /3
861 e 4 π i /3
=(
860
+
+
)
/27,
f 10010 =(
852 e 2 π i /3
872 e 4 π i /3
872
+
+
)
/27,
f 10020
859 e 2 π i /3
879 e 4 π i /3
=(
+
+
)
858
/27,
f 20010 =(
879 e 2 π i /3
859 e 4 π i /3
858
+
+
)
/27,
f 20020
872 e 2 π i /3
852 e 4 π i /3
=(
+
+
)
872
/27.
2 m , where m is an integer and m
In particular, when q
1, it is possible to use the
vector-radix fast Fourier transform (FFT), which is a multidimensional implementation of the
FFT algorithm, for calculating (29) for all
=
a
I A . The complexity of the vector-radix FFT is
q k log q k
O
. In addition, it can be shown that the Yates' Method (Yates, 1937) for efficient
calculation of (19)-(21) in the case of q
(
)
=
2 is equivalent to the vector-radix FFT for calculation
of (29).
4.3 The relation between the Fourier coefficients and the effect of each factor
In a description of experimental design on the basis of an orthonormal system, the model
is expressed by using Fourier coefficients. Fourier coefficients themselves do not provide a
direct representation of the effect of each factor.
On the other hand, since the previous model in experimental design is expressed through the
effect of each factor, it is easy to understand how each factor affects the response variable.
In this section, we present three theorems of the relation between the Fourier coefficients and
the effect of each factor (Ukita & Matsushima, 2011).
First, we present a theorem of the relation between the Fourier coefficient and the general
mean.
Theorem 2. Let ˆ
μ
be the unbiased estimator of the general mean
μ
in the model of Sect.3.1, and let
f 0...0 be that of the Fourier coefficient f 0...0 in the model of Sect.4.1.
Then, the following equation holds:
f 0...0 .
μ =
ˆ
(31)
Next, we present a theorem of the relation between the Fourier coefficients and the effect of
the main factor.
Theorem 3. Let ˆ
in the model of
Sect.3.1, and let f 0...0 a l 0...0 be that of the Fourier coefficient f 0...0 a l 0...0 in the model of Sect.4.1.
Then, the following equation holds:
α l
( ϕ )
be the unbiased estimator of the effect of the main factor
α l
( ϕ )
f 0...0 a l 0...0 .
α l ( ϕ )=
X
( ϕ )
ˆ
(32)
a l
a l
(
)
GF
q
a l =
0
Last, we present a theorem of the relation between the Fourier coefficients and the effect of the
interaction.
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