Performance Evaluation of Image Analysis Methods 399
for the case z < 0, and by using formula [44]
Z
∞
u
x
n
e
−bx
dx = e
−bu
n
X
k=0
n!
k!
·
u
k
b
n−k+1
(u > 0, b > 0), (12.59)
and the mathematical induction method for the case z ≥ 0, we have
h(z) =
Ce
K
0
ˆσ
0
2
z
P
m
0
l=
0
(
l
m
0
)
(m
0
+m
1
−l)!
a
m
0
+m
1
−l+
1
(−z)
l
(z < 0)
Ce
−
K
1
ˆσ
1
2
z
P
m
0
l=
0
(
l
m
0
)
(m
0
+m
1
−l)!
a
m
0
+m
1
−l+
1
(z)
l
(z ≥ 0),
(12.60)
where
(
l
m
0
) =
m
0
!
l!(m
0
−l)!
m
0
=
J−K
0
2
− 1 and m
1
=
J−K
1
2
− 1
a =
K
0
ˆσ
0
+K
1
ˆσ
1
2
C =
1
m
0
!m
1
!
(
K
0
ˆσ
0
2
)
J−K
0
2
(
K
1
ˆσ
1
2
)
J−K
1
2
.
(
12.61)
It has been verified that
R
∞
−∞
h(z)dz = 1.
A.2 Thresholds ∆
1
, ∆
2
,
and ∆
∆
1
, ∆
2
, and ∆ of Eq. (12.14) are functions of J, K
0
, and K
1
. Intensive
simulations
∗∗
showed that they have the following properties:
(1) ∆
1
(K
1
> K
0
) < 0 and ∆
1
(K
1
< K
0
) > 0,
(2) ∆
2
(K
1
6= K
0
) ≤ 0 and monotonically increases with σ
0
,
(3) |∆
2
(K
1
> K
0
)| < |∆
2
(K
1
< K
0
)|,
(4) ∆ < 0 for small and moderate σ
0
and ∆ > 0 only for very large σ
0
.
These findings can be justified as follows.
For (1), it is straightforward to verify using the definition in Eq. (12.14).
For (2), because the frequency of occurrence of |K
1
− K
0
| = 1 is higher
than that of |K
1
− K
0
| > 1, we set |K
1
− K
0
| = 1 in the simulations. In
the case of over-detection (K
1
> K
0
), a new region (say, the l-th region,
∗∗
About the simulation study. A Gibbs Sampler algorithm has been developed (Appendix
9A of Chapter 9) based on the theory in [28]. The algorithm has been implemented on
Macintosh computers with IMSL [57]. Using this algorithm, the various MRF configura-
tions with different resolutions (256×256, 128×128, and 64×64) were generated. The MRF
shown in Figures 12.1 and 12.5 is one of them. Then Gaussian noise with different variances
σ
2
0
= 1, 5, 10i, 100(j + 1), 500(k + 2) (i = 1, · · · , 10, j = 1, · · · , 9, k = 1, · · · , 8) was superim-
posed on each resolution of each MRF configuration. Finally, the iFNM model-based image
analysis method (Chapter 10) was applied to these (over 600) i mages to perform detection,
estimation, and classification.
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