International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
where — A, B, C = extracted features
H4 Hp Hc degree of membership of the features
For the minimum operator the truth value of the result is
defined by the minimum truth value of the used features which
is the logical AND implementation in fuzzy environment.
Similarly, the maximum truth value of all used features
determine the truth value of a class by the maximum operator.
This operator is used in fuzzy as logical OR. For these two
operators, the fuzzy sets of the classes should constitute
complementary membership functions, so the sum of the
degrees of membership for every feature value should be 1.
Therefore, the elements are classified into non-correlated
classes and all features are taken into consideration with the
same importance. In cases where the sum of the degrees of
membership is more than 1, the accordant feature plays a more
important role in the calculation. Using the product operator
this is of lower importance, since only the differences between
the truth values of the classes for a feature cause differences in
the final result. For calculation of a weighted sum, an individual
weight is assigned to each feature. This weight may be constant
to express the reliability of a certain feature in general, but also
variable depending on another feature. For example, the shape
parameter geometry of n longest lines expresses the parallelism
and orthogonality of these lines. However, the reliability of this
feature depends on the size of the object. It can be observed that
this feature provides more reliable values if larger segments are
concerned while at smaller segments only short contour lines
can be extracted which leads — due to noise and rastering effects
- to increasing deviations from parallelism and orthogonality.
The inference procedure results in a crisp value for each
segment and class. In every case the final decision is based on
the maximum method, i.e. the class of highest probability will
be assigned to the corresponding segment. As an example the
confusion matrix for the product operator in shown in Table 1.
In Table 2 the results obtained by different inference operators
are assembled for both test sites. It is obvious that the results are
not independent on the respective operator. Using a
combination of all available features the minimum and
particularly the maximum operator provide results of lower
classification rates. For test site Salem this tendency is more
significant than for Karlsruhe. Product and weighted sum
method achieve higher classification rates of similar dimension.
Other combinations where not all features were included lead to
increasing differences.
Product Buildings Vegetation Terrain
Buildings 95 5 0
Vegetation 4 96 0
Terrain 0 7 93
Table 1. Confusion matrix for the product operator (Salem)
Operator Karlsruhe Salem
Product 90 95
Weighted sum 90 94
Minimum 88 64
Maximum 87 74
Table 2. Classification results by different operators in fuzzy
logic
Figure 10. Classified segments (red- building, blue- terrain,
green- vegetation)
uL ol 55 2 52 5 sse i-2
DE DEI SEE [SE] TE | el ES
eZ BE EBs 5522 22528522
mEIS ele EMSS 8ew solos
en = Aslosiosjo o
+ + + + + 95 96 93 95
+ + + + 93 | 96 | 80 |
+ + + 3: 84 72 87 84
+ + + * 96 88 93 94
+ + + 8S 67 | 73 | $0
+ + + + 93 | 96 | 80 | 92
+ + + 83 96 93 87
zb + + + 89 1 79 | | 38
+ + + 93 | 38 | 93 | 8!
Table 3. Feature combinations for test area Salem
n © 2 29s Q
EE 29 9| o < ss .Cl = 8
EE HE 25 & [EE] "FEE
ew 35x ES ES a= 42924
GER BI HKS SS Sos
en d OB fg 9
+ + zh i 89 90 90
+ + + 93 85 89
+ + zi 86 80 83
+ + + 88 86 87
+ i 91 62 78
+ + 90 89 90
Table 4. Feature combinations for test area Karlsruhe
Due to this quality assessment of different inference operators
product has been selected as standard operator for subsequent
investigations. To compare the reliability of the defined features
and to demonstrate the influence of each of them 9 different
feature combinations have been calculated and the influence of
missing features has been observed, whereas the independence
of the features was assumed. These feature combinations and
their results can be seen in Table 3 and 4. Besides the individual
class-related values also an overall classification rate has been
included. The results show that the amount of significant border
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