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the type of outliers, the number of outliers and the posi-
tion of outliers added to the mixed distribution. We can
distinguish two main types of outliers, the ones that follow
a certain pattern e.g. a cluster (coherent outliers) and those
that are positioned in random places (random outliers). The
influence of the outliers depends on their distance from the
mixed distribution, so various distances will be examined to
demonstrate how they will affect the obtained results. We
will examine how many outliers the method can tolerate.
The performance of the Hough transform method will be
compared with the solution obtained by the least square error
method, where the mean of each distribution is computed and
used as a mixed pixel or as an “endmember” in the classical
unmixing approach.
3.1 Coherent outliers.
The outliers of this type tend to create clusters in an arbitrary
distance from the mixed distribution. Such outliers are shown
in Figure 2. Outliers that are placed too close to the distri-
bution do not create much, if any, distortion to the obtained
results, while outliers placed too far away are very unlikely to
be present in real applications.
9o
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Figure 2: Outliers placed on a given distance (3, 6 and 9
times the standard deviation) from the mixed set.
However, distant outliers may be present in the mixture dis-
tribution due to the existence of another class in the distri-
bution, that we have no data to describe it. At first we will
consider a fixed amount of outliers (e.g. 10% of the given
set) and we will place the cluster of those outliers in different
distances from the mixed distribution. The unit used to ex-
press these distances is the standard deviation of the mixed
distribution, which in these experiments is rather tight. The
standard deviation of the cluster of outliers will be half the
standard deviation of the mixed set. Then we will examine
different number of outliers placed at various distances. The
results obtained from this experiment are shown in Table 1.
From Table 1 we can see that estimates of the mixing propor-
tions of the Hough method are not affected by the outliers.
87
Out Hough
| |
| a(%) | b(75) | c(%) |
| Dist
LSE )
La) 150) Le)
1 5
58 10
54 1
Table 1: Effect of number of outliers in the mixed distribution.
Mixed set composition a = 30%, b = 60%, c = 10%.
The LSE method though seems to be affected, as it was ex-
pected, and its performance deteriorates as the outliers were
placed further away and increase in numbers.
If the cluster of outliers is within the convex hull defined by
the “pure” class sets, then these outliers may indicate that
the mixed set is not actually homogeneous as assumed so far,
but patchy and the outliers in this case represent a patch of
another mixed area with different composition. In this case
it would be interesting to be able to identify the two mixture
compositions. In order to achieve this, we will examine the
second significant peak in the Hough space as well.
For the next experiment we created a testing set that is com-
prised of two mixtures with different compositions. If the
mixture of outliers had similar composition to the one of the
mixed set, then it would have been very difficult to distin-
guish between them. That is why for the outliers mixture
only, compositions with different dominant class were exam-
ined. The original mixed set still had composition a = 30%,
b — 60%, c — 10% as in the previous experiments. The re-
sults obtained can be seen in Table 2. For this experiment
1/3 of the test set belongs to the outlier mixture. The second
peak in the Hough space is also examined.
Outliers Mix || Hough I|
| First Peak | Second Peak ||
32-60-8 17- 5 1-44-25
- -24
32-60-8 16-26
LSE
30-10-60
60-30-1 41-51-8
Table 2: Test set comprised of two mixtures 2/3 from a
mixture with composition (30% — 60% — 10%) and the other
1/3 (outliers) with varied composition.
As we can see in Table 2 the other peaks in the Hough space
may be used to identify sets of coherent outliers. Suppose
that we used only one band for the proportion estimation. If
the distribution of the mixed set is not very coarse in compar-
ison to the bin size used to discritized the Hough space, then
the corresponding points in the accumulator tend to confine
in a line. If we use in addition a second band, as is the case
here, the points of the mixed set in the second band tend to
give a different line that crosses the first line and the crossing
point will be the answer. These two lines are not equally im-
portant, depending on how separable are the classes in each
band and of course on the variance of the mixed set in each
band.
When we examine two different mixtures at the same time,
then we will generally expect to see four different lines that
intersect in four different points. For this case we visualise
the Hough space as shown in Figure 3 to check what is going
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B7. Vienna 1996
