on. If the mixtures have the same number of points then all
four intersection points are equally probable to be the right
answer and the result will depend on the position of each
mixture and the shape of that set. If the two mixtures are
not very separable then the corresponding lines will be very
close to each other and the corresponding crossing points
may be reduced to three, two or even one. In these cases
the mixtures are almost identical. If one of the mixtures has
fewer points than the other, the lines generated by it will be
degraded and the dominant mixture should be responsible for
the outcome.
Figure 3: The mixed set contains points from two mixtures:
half of the points belong to a mixture with composition 30-60-
10 and the other half to a mixture with composition 30-10-60.
The lines were added to show the lines in the Hough space
of each mixture in each band. The four numbers indicate the
four crossing points and correspond to the following mixtures:
1 — (27-16-47), 2 — (12-49-39), 3 — (27-60-13), 4 — (47-
27-26)
3.2 Random outliers
This type of outliers does not form a coherent set and their
distance from the mixed set is randomly chosen in the range
between 0 and 12 standard deviations. Such outliers can be
seen in Figure 4.
In this experiment, for a certain number of random outliers, a
number of mixed sets were generated and tested. The error in
proportion estimation was calculated, and finally the average
and the standard deviation of the errors (given in brackets)
in proportion estimation based on 100 experiments are shown
in Table 3. The error in estimation of each proportion (i.e
proportion a) was calculated as errora = 100 x la—27l where
a is the estimated value for a and ar is the true value of a.
As we can see in Table3 the Hough method performs very well
and remains remarkably stable throughout the experiment.
The LSE method seems to be affected by the outliers and its
performance vary depending on the position of the outliers.
Ironically, the more the outliers are and the more uniformly
distributed about the mixed distribution they are, the bet-
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Table 3: Effect of random outliers in the mixed distribution.
Mixed set composition a = 30%, b = 60%, c = 10%.
ter the LSE method will perform because the mean of the
mixture distribution will not be affected by their presence.
However, such an improvement in the performance of the
classical method is clearly artificial.
4 WHAT IF SITES OF PURE CLASSES ARE NOT
AVAILABLE?
Ideally, for pure classes we would like to use sets of pixels
representative of the pure classes extracted from the remotely
sensed image itself. However sometimes, especially if the
terrain tends to vary at smaller scales than the size of the
test sites, it is difficult to find homogeneous test sites that
belong solely to a given pure class.
A solution to this problem is to derive the attributes of the
pure classes from test sites for which ground measurements
are available [Pech et al., 1986] [Gong et al., 1994]. Accord-
ing to our model we have:
Ww pd b5y4os (4)
We can make use of the Hough transform again to identify
the best values for z, y and z, if we consider that equation
4 is an equation of a plane in the 3D space (a, b, c), which
is parameterised by different values of w and we are inter-
ested in identifying the luminances z, y, z. In this case we
have a 3D accumulator array defined in the parametric (z,
y, z) domain. We can then use the luminance values w of
the training sites, with the estimated (by ground inspection)
values of their mixture parameters, to identify values of (x,
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B7. Vienna 1996
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