404 Statistics of Medical Imaging
where
J
k
= Number of x
j
∈ R
k
ˆµ =
1
J
P
J
j=
1
x
j
ˆµ
k
=
1
J
k
P
J
k
j=
1
x
j
J =
P
K
k=1
J
k
.
(12.77)
Because
K
X
k=1
J
k
(ˆµ
k
− ˆµ)
2
=
K
X
k=1
J
k
ˆµ
k
2
− J ˆµ
2
, (12.78)
and (from Eq. (12.21)),
K
X
k=1
J
k
X
j=1
(x
j
− ˆµ
k
)
2
= J ˆσ
2
, (12.79)
Eq. (12.76) becomes
K
X
k=1
J
k
X
j=1
(x
j
− ˆµ)
2
+ J ˆµ
2
=
K
X
k=1
J
k
ˆµ
k
2
+ J ˆσ
2
. (12.80)
For a given image,
P
K
k=1
P
J
k
j=1
(x
j
− ˆµ)
2
+ J ˆµ
2
is constant. Thus, we have
K
1
X
k=1
J
k
ˆµ
k
2
+ J ˆσ
2
1
=
K
0
X
k=1
J
k
ˆµ
k
2
+ J ˆσ
2
0
. (12.81)
Eq. (12.81) leads to Eq. (12.2 2).
Problems
12.1. Prove the properties of Eq. (12.2 9).
12.2. Derive Eq. (12.37).
12.3. Derive Eq. (12.41).
12.4. Derive Eq. (12.46).
12.5. The mathematical induction method is us ed in the derivation of
Eq. (12.60). Show the details of this mathematical induction.
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