Figure 3: the regions boundaries of Fig. 1
and the variance 0 = 5 is added to each pixel of the im-
age, and the maximal difference of the noise is 40. With the
risk error a = 0.1% both function models (27) and (28) are
tested. The segmented results of the planar function model
(27) are shown in the Figure 2. From the figure 2 we can
see that the lower-right part of the large circle is segmented
as an independent region, because this part is isolated from
the two other parts by the triangle. The segmentation here
is entirely correct. The boundaries of the segmented regions
are shown in the figure 3. The results of the horizontal plane
function model is same, except that there is a very small un-
necessary region on the boundary between the triangle and
the small circle, which is caused by the large noise and can
be easily eliminated.
Figure 4: the second image with e — 8
Figure 4 is also a simulated image with normally distributed
errors. It has 8 regions and the difference of the avarage
grayvalues of two neighbouring regions is about 32. The
variance of the Gaussian noise is à = 8, and the maximal
difference of the noise is about 65(+40). That means the
grayvalues between two neighbouring regions have already
overlapped. We have also tested two function models for
the image. By risk error a = 0.1% both models can only
segment the image into three regions, because the regions
can not separated in this case. By a = 1% both the models
can divide the image correctly except that there is a small
Figure 5: the segmented results of Fig. 4
Figure 6: the regions boundaries of Fig. 4
unnecessary region as in the first image. From the equations
(43) and (44) we know that by larger a the separability of
the regions increases too. The segmented results and the
boundaries are shown respectively in the figure 5 and 6.
A
Fig. 7 the 3. image with o = 20 Fig. 8 the coarse segmentation
ja
Fig. 9 results after region-merging Fig. 10 regions boundaries of Fig. 7
m
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