380 Statistics of Medical Imaging
−200 −100 0 100 200
0
0.2
0.4
0.6
0.8
FIGURE 12.2
The pdfs of over-detection in the case where σ
2
0
= 10, 40 , 70 (curves from
left to right).
−300 −200 −100 0 100 200 300
0
0.2
0.4
0.6
0.8
FIGURE 12.3
The pdfs of under-detection in the case σ
2
0
= 10, 40, 70 (cur ves from left to
right).
Z
′
is ∆
′
=
∆−µ
σ
. Thus, the error-detection probability, P
e
= P (Z < ∆)
of Eq. (12.15), can be approximated by the cumulative distribution function
(cdf) of Z
′
at Z
′
= ∆
′
, that is, Φ(∆
′
) of Eq . (12.51). For ima ges in Figures 12 .1.
a − l, Φ(∆
′
) are given in Table 12.4. Bec ause Φ(∆
′
) (cdf of the standard
Gaussian) monotonica lly increases with ∆
′
, Table 12.4 shows that the over-
detection pro bability decreases and the under-detection probability increases
as the variance o f the image increases.
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