Performance Evaluation of Image Analysis Methods 393
TABLE 12.10
The Relative Errors a nd Estimation Interva ls of the Parameter
Estimates of the Images in Figures 12.1a–l
ˆπ
k
F
igures 12.1a–f SN R ≥ 14.2 db ε
π
< 0.4% ˆπ
k
∈ ̟
ˆπ
k
ˆµ
k
F
igures 12.1–h SN R ≥ 12.4 db ε
µ
< 0.6% ˆµ
k
∈ ̟
ˆµ
k
ˆσ
2
k
F
igures 12.1a–f SN R ≥ 14.2 db ε
σ
< 5.0% ˆσ
2
k
∈ ̟
ˆσ
2
k
TABLE 12.11
Estimation Results (Images in Figures 12.4.a)
k ˆπ
k
ˆµ
k
ˆσ
2
k
CRLB
ˆπ
k
C
RLB
ˆµ
k
CRLB
ˆσ
2
k
× 10
6
1 0.2348 −634.6 56271 0.00044 160 .4 6.020
2 0.2167 −122.5 125 0.00038 6.6 0.008
3 0.2460 58.9 488 0.00041 1.5 0.001
4 0.1467 815.5 307284 0.00026 470.4 26.11
5 0.1558 1290.2 62701 0.00028 116 4.5 170.9
= ˆπ
k
0
K
0
X
k=
1,k6=k
0
(Φ(
x
′′
k
− ˆµ
k
0
ˆσ
k
0
) − Φ
(
x
′
k
− ˆµ
k
0
ˆσ
k
0
)
), (12.50)
where •|k
0
represents the event that x comes from R
k
0
but is classifie d (by
the classifier) as coming from R
k
(k 6= k
0
): x
′
k
< x ≤ x
′′
k
, and Φ(y) is cdf of
the standard Gaussian random variable given by
Φ(y) =
1
√
2π
Z
y
−∞
e
−
x
2
2
dx. (12.51)
12.2.3.1 Results of Classification Performance
1) Results from simulated images
The simulated images shown in Figure 12.1 are used here for evaluating
classification performance. Image analysis is implemented using the iFNM
model-based image analysis method of Chapter 10. The resultant imag es are
shown in Figure 12.5, with the same labelings as for Figure 12.1. Image re-
gions are represented by the mean values of pixels in each imag e region. In
Figure 12.5, four grAy levels (white, light gray, dark grey, and black) are used
for the four image regions.
Comparing these images with their counterparts in Fig ure 12.1 shows that,
for the images in Figure 12.5a–12.5d, there is almos t no classification error; for
the imag es in Figure 12.5e–12.5f, there is a very small amount of error; for the
images in Figure 12.5g–12.5l, there are some errors. These e rrors are mainly
due to isolated pixe ls inside one image region being mis classified into another
image region. The region shapes, however, are all preserved. This type of er-
ror occurs due to the fact that the Bayesian criterion (Eq. (12.49)) actually
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