376 Statistics of Medical Imaging
where
(
l
m
) =
m!
l!(m−l)!
m =
J−K
0
2
− 1 or m =
J−K
1
2
− 1
a =
K
0
ˆσ
0
+K
1
ˆσ
1
2
C =
1
(m!)
2
(
K
0
ˆσ
0
K
1
ˆσ
1
4
)
m+
1
.
(12.17)
P
ov
and P
ud
are given by
P
ov
= P
ud
=
m
X
l=0
(
m
2m−l
)
γ
l+1
e
0
(γ + γ
−1
)
2m−l+
1
(∆ < 0), (12.18)
P
ud
=
m
X
l=0
(
m
2m−l
)
(γ
l+1
+ γ
−l−1
(1 − e
1
))
(γ + γ
−1
)
2m−l+
‘1
(∆ ≥ 0), (12.19)
where
γ =
r
K
1
ˆσ
1
K
0
ˆσ
0
e
0
=
P
l
j=
0
(−
K
0
ˆσ
0
2
∆)
j
j!
e
K
0
ˆσ
0
2
∆
e
1
=
P
l
j=
0
(
K
1
ˆσ
1
2
∆)
j
j!
e
−
K
1
ˆσ
1
2
∆
.
(
12.20)
Although P
ov
and P
ud
have the same functional form in the case that ∆ < 0
(see Eq. (12.18)), their values are different. This is because the parameters
used in Eq. (12.18), for example, K
1
and ˆσ
1
, have differ ent values in the over-
and under-detection cases, respectively. ∆ > 0 occ urs only in the case of
under-detection: either ˆσ is very large or γ = 1, w hich leads to ∆
2
= 0. From
Eqs. (12.18) through (12.20), it is clear that the error-detection probabilities
of the number of image regions, P
ov
and P
ud
, are functions of the number of
pixels (J), the number of image regions (K
0
, K
1
), and the variances of the
image (ˆσ
0
, ˆσ
1
).
12.2.1.2 Error-Detection Probabili ties and Image Q ual ity
When the variance of the entire image is defined by
ˆσ
2
=
K
X
k=1
ˆπ
k
ˆσ
2
k
, (12.21)
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