clusters. A small region is grouped with its most similar
neighbour expressed by the t-score according Eq.( 12).
4.3.3 Stage IIc: Removal of Insignificant Regions
The aim of Stage IIc is to remove elongated regions which
are due to mixed pixels at the region borders. Removal of
regions solely based on size is insufficient to perform this
task. In Lemmens (1991) it is shown that an appropriate
measure to describe the significance of a sliver polygon is
the quotient of the area size A of the region and the stan-
dard deviation of the area c4, which is a z-score. Accord-
ing a one-sided z-test a 97.596 confidence leads to the test
statistic: A < 1.96 04, to accept the assumption that the
region is insignificant. The size of a region is here simply
the number of pixels.
(Ibid) further shows that if we may assume that all co-
ordinates are uncorrelated, than:
n
o2 702 Ga — yai)? (zia — Sy (13)
i=]
where ((zi,y;),? = 1,..,n) are the coordinates of the
border pixels of the region and 9? the variance of the co-
ordinates of the border pixels, which should be known a
priori. In Stage IIc we have implemented the above ap-
proach. It has to be emphasized that the variance of the
coordinates should not be interpreted here as a physical
meaningful measure, but as a measure that expresses the
desired minimal extension of the regions.
5 Implementation Considerations
5.1 A Feasible Computer Implementation
Here we treat an implementation that uses no a priori
knowledge about initial regions. We start examining the
image in one of its corners, in particular the left-upper cor-
ner, but any other corner would be appropriate. Conse-
quently the left-upper corner pixel (1,1) is the first, initial
region, receiving label 1. So, we have to predict 41(1,2)
from g(1,1) according Eq.( 4). If the prediction g1(1,2) is
sufficiently close to the actual value g(1, 2), then pixel (1.2),
receives label 1, else it receives label 2. Suppose label 1 is
assigned to pixel (1,2). Next, pixel (1,3) is predicted from
pixels (1, 1) and (1,2). In this way the predictor moves over
the image with step size of one pixel.
=
|
1 | | |
i-4 i-3 i-2 i - 1 |
Fig. 1 Sampling and truncation of the predictor r(p,q) —
ezp[— ((p?4-q?)/2.?)1/] The function is truncated after i—3,
since there the weights are becoming insignificantly small, g;
1s estimated from g;_1, g;_, and gi-1, according the sampled
weights.
796
Hn ——Á— n
uae .0067 |.0111 mt .0015 ns
.0015/.0183/.0821 |.1353 |.0821 |.0183 |.001
.0067|.0821|.3679 |.6065 |.3679 |.0821 —
gm -
= .0111 |
mom mI
!
Fig. 2 Kernel of the predictor for w — 1.
The prediction elements (p,q) of Eq.( 4), can be pre-
computed and stored in a half plane kernel, as shown in
fig. 2. The kernel is truncated if its elements are becoming
insignificant small, e.g. Etrune = 0.01, which is 1% of the
maximum value of r(p,q) : r(0,0) = 1. The values of the
elements of the predictor are entirely determined by w and
the predictor size by Etrune Note that the kernel in fig. 2 is
just an example.
w < 1 yields a small predictor, making the prediction
sensitive to local grey value anomalies. The grey value of
the same cbject may change gradually, when moving from
the one side to the other side. A large predictor is not able
to handle such gradual grey value changes. Our experi-
ments showed that the value of w may vary freely whitin
the range [1.5 - 5], without affecting the final segmentation.
To remove phantom, small and insignificant regions pro-
duced by Stage I, in Stage II iteratively two regions are
compared on similarity, according the t-score Eq.( 10). The
merging of two regions, affects the statistical properties of
the joined region and hence the t-score of the new region
and its adjacent regions. To make the merging process or-
der independent, first the t-scores of all adjacent regions
are determined. Next the two regions which have the low-
est t-score are merged. The statistics of the new region are
computed and for all the former neighbours of the former
two regions t-scores are computed. Next the table is traced
from the beginning to the end again to find the smallest
t-score. This process is repeated until no t-score exceeds
the critical value anymore.
5.2 Example
We demonstrate our procedure by an example. Fig. 3 shows
a scene with six regions. This scene is recorded as a 162
image in the grey value range [0 — 100]. Fig. 4 shows a
part of it. Additionally, for orientation purposes the region
boundaries are drawn and the predictor is superimposed to
predict pixel (8, 11).
5.2.1 Stage I: The Prediction Stage
The four adjacent pixels (8, 10), (7, 10), (7, 11) and (7, 12),
all are part of different regions, 8,5,6 and 3 respectively.
The variance of the noise is estimated from the rectangle
with corners (2, 11) and (6, 15) yielding o, — 2.26. The pre-
diction equation Eq.( 4) results in the figures summarized
in table. 1.
k | 9) (8,11) XQ) |z-score Ho
8 9.75 0.8196 | 1.21 | accepted
5 76.71 0.7967 | 36.49 | rejected
6
3
89. 0.8532 | 39.93 | rejected
80.15 0.8243 | 37.12 | rejected
Table 1
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