n than
iS
(33)
test:
(37)
Int).
- ] line
(38)
dized
(39)
first n
se the
(40)
(41)
a risk
lity of
‚ given
m the
Jue of
o, the
pothe-
of the
There
lity of
equal
ton a
ude of
model
he hy-
of test
B? It is the problem of separability of regions. If the pixel
Pn+1 does not belong to the region Rj but to the region
R;+1, only if the alternative hypothesis (35) is accepted,
the region R;+1 can be separated from the region Rj. For
a given « and f, we can obtain a minimal uncenterized
parameter 60,
60 = 6(a, B) (42)
For the standardized normal distribution óg is
60 — Ki-s + Kg (43)
For example, if a = 0.1%, 8 — 8096, then 60 = 4.13. If the
value of the statistic variable w; is greater or equal to the
60, then the model error can be found with the given a and
B. So the minimal model error should be
0 0 bo
: V vivi
That means, if and only if the difference of the grayvalues
between the region P and R41 should be equal or greater
than s?, the two regions can be separated from each other
with the given risk error o and power of test B.
(44)
If the number of the observations n is great enough, the
du 2 1, so in this case the equation (44) can be approx-
imated as:
5} = 060 (45)
4. IMPLEMETATION OF SEGMENTATION
4.1 Choice of Start Area
For the different function models of à region we must choose
a start area with certain number of pixels, e.g. for the
planar function (27) the start area should have three pixels.
We use the procedure of minimal difference in grayvalues
(MDG) to set up the start area:
e to find an unclassified pixel as the first one;
e to choose the pixel with MDG to the first pixel among
all the neighbouring pixels;
e the next pixel should have MDG to the average gray-
value of the choosen pixels.
The neighbouring pixel can stay in the 4 directions or 8
directions. We should pay attention to the case when all
the pixels in the start area are in the same line. In this
case the solution of the equation (31) are indefinite. As an
alternative we can then use the function model (28).
How to choose the first pixel is also quite important. It
may have better results if the first pixel does not stay on
the boundary of two regions.
4.2 Labelling of Regions
In order to keep the implementation as quick as possible,
the labelling of the pixels should be optimized. The unde-
tected pixels, the rejected pixels and the accepted pixels can
be separatly labelled in order to avoid repeatedly searching.
By searching for the acceptable pixels the rules about pri-
ority of the nearest neighbouring pixel (lengthwise priority)
and the shortest distance to the start pixel (crosswise pri-
ority) can be used. After a region has been segmented, all
the pixels in the region will be also identified and specially
labelled.
After all the regions have been segmented, the boundaries
of them can be also easily obtained.
4.3 Handling the Small Areas
If two regions have a great difference in grayvalues, then
the grayvalues between two sides of the boundary do not
change suddenly, but gradually. So sometimes there may
be a small region along the boundary found. These kinds
of small regions are usually very narrow and superfluous.
According to their properties we can merge them. Here we
can also use the previous knowledge if there is any.
4.4 Some Examples
In order to examine the effectiveness of the method we have
carried out several experiments. The results of them are
illustrated in the following figures.
Figure 1: the first image with c — 5
Figure 2: the segmented results of Fig. 1
Figure 1 is a simulated image with 5 regions. The range
of the grayvalues of the image is from 0 to 255. The differ-
ence of the average grayvalues of two neighbouring regions is
about 50. A Gaussian noise with the expectation E(e) = 0
