Performance Evaluation of Image Analysis Methods 377
then, using the theory of Analysis of Variance [46], Appendix 12B shows that,
when the image is partitioned into K
0
and K
1
image regions, we have
ˆσ
2
1
(1 +
K
1
X
k=1
ˆπ
k
ˆµ
2
k
ˆσ
2
1
)
= ˆσ
2
0
(1 +
K
0
X
k=1
ˆπ
k
ˆµ
2
k
ˆσ
2
0
). (
12.22)
Define the signal-to-noise ratios SN R
k
and SNR of the k-th image region
and of the entire image, r espectively, by
SNR
k
=
ˆµ
2
k
ˆσ
2
and SN R =
K
X
k=1
ˆπ
k
SNR
k
. (12.23)
Then Eq. (12.22) becomes
ˆσ
2
1
(1 + SNR(K
1
)) = ˆσ
2
0
(1 + SNR(K
0
)), (12.24)
where SN R(K
0
) and SNR(K
1
) are the SN Rs of the partitioned ima ges with
K
0
and K
1
image regions, respectively. Thus, γ in Eqs. (12.18) through (12.20)
actually is
γ
2
=
K
1
K
0
s
1 + SNR(K
0
)
1 + SNR(K
1
)
. (12.25)
Eqs. (12.18), (12.19), and (12.25) show that the error-detection probability
P
ov
and P
ud
are functions of SNR, J, K
0
, and K
1
. SN R is a meas ure of image
quality; J is an indicator of image resolution (for a fixed field of view); K
0
and K
1
are indicators of image complexity. Thus, error-detection probabilities
depe nd on image quality, resolution, and complexity. This is a very sens ible
result and establishes a theoretical guideline for e rror de tec tion in any image
whose intensities follow the iFNM model.
12.2.1.3 Results of Detection Performance
1) Results from simulated images
Figure 12.1 shows twelve simulated images labeled a to l. As mentioned
earlier (in the footnote), the regions o f ea ch image are gener ated by a Gibbs
sampler [28, 33] and have b een corrupted by Gaussian noise. The settings and
the variances σ
2
0
(as well as the signal-to-noise ratios SNR) of these images
are given in Tables 12.1 and 12.2, respectively
†
.
The detection results are summarized in Table 12.3. It shows that the num-
ber of image regions indicated by MDL criterion is correct for all 12 images:
K
1
= Arg{min
1<K<7
I
K
} = 4 is equal to the corr ect number of image regions,
K
0
= 4. This result demons trates that the MDL criterion has a very robust
†
The purp ose of making the region means take on values from the negative to the positive is
to simulate X-ray CT i mages. SN R (db) = 10 log
10
(SNR), SNR is defined by Eq. (12.23).
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