Image Processing Reference
In-Depth Information
Theorem 4. Let ˆ
β l , m
( ϕ
,
ψ )
be the unbiased estimator of the effect of the interaction
β l , m
( ϕ
,
ψ )
in the
f 0...0 a l 0...0 a m 0...0 be that of the Fourier coefficient f 0...0 a l 0...0 a m 0...0 in the model
model of Sect.3.1, and let
of Sect.4.1.
Then, the following equation holds:
ˆ
f 0...0 a l 0...0 a m 0...0 .
ψ )=
a l
a m
β l , m ( ϕ
,
X
( ϕ ) X
( ψ )
(33)
a l
a m
GF
(
q
)
GF
(
q
)
a l =
a m =
0
0
From these theorems, the effect of each factor can be easily obtained from the computed
Fourier coefficients.
=
=
Example 11. Let q
5 . Consider the general mean, the effect of main factor F 1 , and the
effect of the interaction of F 1 and F 2 . Then,
3 and n
e 2 π ilk /3 .
X
(
k
)=
(34)
l
f 00000 holds.
Next, using (32) and (34), the following equations
μ =
First, using (31), ˆ
f 10000 +
f 20000 ,
ˆ
α 1 (
0
)=
e 2 π i /3 f 10000
e 4 π i /3 f 20000 ,
(
)=
+
ˆ
α 1
1
e 4 π i /3 f 10000
e 2 π i /3 f 20000 ,
ˆ
α 1
(
2
)=
+
hold. Hence, it is clear that the effects of main factor F 1 ( 3 parameters) can be obtained from the
computed Fourier coefficients ( 2 parameters).
Last, using (33) and (34), the following equations
f 11000
f 12000
f 21000
f 22000 ,
ˆ
(
)=
+
+
+
β 1,2
0, 0
ˆ
e 2 π i /3 f 11000 +
e 4 π i /3 f 12000 +
e 2 π i /3 f 21000 +
e 4 π i /3 f 22000 ,
β 1,2 (
0, 1
)=
e 4 π i /3 f 11000
e 2 π i /3 f 12000
e 4 π i /3 f 21000
e 2 π i /3 f 22000 ,
ˆ
(
)=
+
+
+
β 1,2
0, 2
ˆ
e 2 π i /3 f 11000 +
e 2 π i /3 f 12000 +
e 4 π i /3 f 21000 +
e 4 π i /3 f 22000 ,
β 1,2 (
1, 0
)=
ˆ
e 4 π i /3 f 11000
f 12000
f 21000
e 2 π i /3 f 22000 ,
(
)=
+
+
+
β 1,2
1, 1
ˆ
f 11000 +
e 4 π i /3 f 12000 +
e 2 π i /3 f 21000 +
f 22000 ,
β 1,2 (
1, 2
)=
e 4 π i /3 f 11000
e 4 π i /3 f 12000
e 2 π i /3 f 21000
e 2 π i /3 f 22000 ,
ˆ
(
)=
+
+
+
β 1,2
2, 0
ˆ
f 11000
e 2 π i /3 f 12000
e 4 π i /3 f 21000
f 22000 ,
β 1,2
(
2, 1
)=
+
+
+
ˆ
e 2 π i /3 f 11000 +
f 12000 +
f 21000 +
e 4 π i /3 f 22000 ,
β 1,2 (
2, 2
)=
hold. Hence, it is clear that the effects of the interaction of F 1 and F 2 ( 9 parameters) can be obtained
from the computed Fourier coefficients ( 4 parameters).
From these theorems, the effect of each factor can be easily obtained from the Fourier
coefficients. Therefore, it is possible to implement easily the estimation procedures as well
as to understand how each factor affects the response variable in a model based on an
orthonormal system.
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