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Various experiments have been carried out to test the performance
of the HMRFEM framework. An example is shown in the following
figures. Figure 2(a) shows a simulated 3class image
sampled from an MRF model using the Gibbs sampler. The intensities
for the three classes are 30, 125 and 220 respectively. Figure
2(b)(e) show the same images with added Gaussian noise
with standard deviation of 28, 47, 66, and 95. Because image
contrast is what we are most interested in for examining qualities
of an image, a measurement of the noise is more meaningful with
image contrast being taken into account. Thus we define a measure,
the noisetocontrast ratio (NCR) as the following:
Thus, the NCRs of the four test images are 0.3, 0.5, 0.7 and 1.0,
respectively. Figure 2(f)(k) show their intensity
histograms. Except for the first, each histogram exhibits severe
overlap. The true parameters for the test images are listed in
Table 1.
Figure 2:
Test images for parameter estimation. (a) the original
image; (b)(e) noisy images with NCR 0.3, 0.5, 0.7, and 1.0;
(f)(k) histogram of (b)(e).

Table 1:
True model parameters of Figure
2(b)(e).

Table 2:
Initial parameter estimation using discriminant
measurebased thresholding.

The discriminant measurebased thresholding method is then applied
to each of the four test images to estimate the initial
parameters. Table 2 shows the results. Comparing it
with Table 1, we can see that the estimates are
acceptable, especially when the noise level is low.
The standard FMEM algorithm and the HMRFEM algorithm are then
applied to the four test images until there is no significant
change in the value of the Qfunction. To measure the
segmentation accuracy, we also define the misclassification ratio
(MCR), which is
The standard FMEM algorithm only converges for the first image,
which has the lowest noise level (NCR=0.3). In this case, the
estimation results and the number of iterations K are shown in
Table 3. With those estimated parameters, we reconstruct
the histogram and obtain the segmentation, as shown in Figure
3. Note that, the parameter estimation is not accurate
when compared with their true values listed in Table 1.
Table 3:
Parameter estimation using the FMEM algorithm

Figure 3:
Parameter estimation for Figure 2(b) using the
standard FMEM algorithm. (a) the reconstructed histogram; (b) the
segmentation with MCR 5.82%.

The HMRFEM algorithm rapidly converges for all the four test
images. Table 4 and Figure 4 show the results.
Taking the true parameters shown in Table 1 as the
references and comparing the results from the two methods, it can
be seen that: (1) the HMRFEM algorithm gives more accurate
estimates for parameters; (2) the HMRFEM algorithm provides
automatic segmentation with much lower MCR.
Table 4:
Parameter estimation using the HMRFEM algorithm

Figure 4:
Parameter estimation for Figure 2(b)(e) using
the HMRFEM algorithm. top row: the reconstructed histograms;
bottom row: the segmentations.

Next: Segmentation of Brain MR
Up: Segmentation Using the HMRFEM
Previous: Initial Parameter Estimation
Yongyue Zhang
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