Performance Evaluation of Image Analysis Methods 391
12.9. Eq. (12.37) shows tha t the Cramer –Rao L ow Bounds of the variances of
weight estimates do not depend o n the variance σ
2
0
of the image. Eqs. (12.41)
and (12.46) show that the Cramer–Rao Low Bounds of the variances of mean
and variance estimates depend on the variance σ
2
0
of the image and increase
as σ
2
0
increases. These facts a re a lso reflected in Table 12.9.
The accuracy of the estimation can be evaluated by judging if the re lative
error s are small and the estimates ˆπ
k
, ˆµ
k
, and ˆσ
2
k
fall into these estimation
intervals. Using Tables 12.1 and 12.9, the results are summarized in Table
12.10, which shows that for the images with SN R > 14.2 db, the relative
error s of the weight and the mean are less than 0.6%; all parameter estimates
are in the Cramer -Rao estimation intervals.
¶
2) Results from the real image
For the real image shown in Figure 12.4a, the ground truth is given in Table
12.5. The parameter estimates and Cramer–Ra o Low Bounds of the variances
of these estimates are given in Table 12.11. Using Tables 12 .5 and 12.11 , we
note that the relative erro rs of the weights and the means (k = 2, ···, 5) are
less than 8.4% and 2.1 %, respectively, ˆπ
k
∈ ̟
ˆπ
k
(k = 2, ···, 5), ˆµ
k
∈ ̟
ˆµ
k
(k = 2 , ···, 5), and ˆσ
2
k
∈ ̟
ˆσ
2
k
(k = 5).
k
For the imag e shown in Figure 12.1h (its SNR = 12.4 db is similar to
SNR = 12.6 db of this real image), ˆπ
k
∈ ̟
ˆπ
k
(k = 1, 2, 4), ˆµ
k
∈ ̟
ˆµ
k
(k =
1, 2, 3, 4). These results show that the estimation performanc e for the images
in Figures 12.1h and 12.4a are very simila r.
12.2.3 Classification Performance
Let IMG(J, K) denote an image of J pixels (x
j
, j = 1, ···, J) and K image
regions (R
k
, k = 1, ···, K). The iFNM model-based image analysis method
(Chapter 10) uses a Bayesian classifier to classify pixels into image regio ns.
The decision rule in pixel classification is
x
j
∈ R
k
0
if ˆπ
k
0
g(x|
ˆ
θ
k
0
) > ˆπ
k
g(x|
ˆ
θ
k
) (k = 1, ···, K
0
, k 6= k
0
). (12.49)
Suppo se an image is partitioned into K
0
image regions, R
1
, ···, R
K
0
. Then
the probability of misclassification, given that the true image region is R
k
0
,
is [50]
P
mis
(•|k
0
) = ˆπ
k
0
K
0
X
k=1,k6=k
0
Z
R
k
g(x|
ˆ
θ
k
0
)dx
¶
Only the variances of the 2nd image region of images in Figures 12.1.a - 12.1.c and 12.1.f
are slightly outside ̟
ˆσ
2
2
.
k
Here we consider only the second through fifth image regions (i.e. , Poly, 013A, Teflon, and
Bone), because the first im age region (rings) does not represent any real physical object,
and is a product of image reconstruction.
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