In: Paparoditis N., Pierrot-Deseilligny M.. Mallet C. Tournaire O. (Eds), IAPRS. Vol. XXXV1I1. Part ЗА - Saint-Mandé, France. September 1-3. 2010 
multiple classes is hierarchically subdivided progressively into 
more homogeneous clusters using a binary splitting rule to form 
the tree, which is then used to classify other similar datasets. 
CTs have the advantage that they also work when the 
classification variables are a mixture of categorical and 
continuous. In the final classification not all but only the most 
prominent attributes are used. This makes the classification 
method highly automatic and different from most other 
approaches in which the input data must remain fixed. The 
Entropy model was used as the splitting criteria in our study. 
Also, the trees were pruned through a 10-fold cross validation 
process, which has been demonstrated to produce highly 
accurate results without requiring an independent dataset for 
assessing the accuracy of the model. In the original output of 
the CTs, each pixel is associated w'ith a degree of membership 
for the class at which particular leaf it was classified. If a pixel 
is not associated with that class, it will be assigned a zero. 
4.4 Fuzzy Majority’ Voting Based Combination 
The idea is to give some semantics or meaning to the weights. 
Therefore, based on these semantics the values for the weights 
can be provided directly. In the following the semantics based 
on fuzzy linguistic quantifiers for the weights are used. The 
fuzzy linguistic quantifiers w'ere introduced by Zadeh (1983). 
w'ho defined two basic types of quantifiers: absolute, and 
relative. Here the focus is on relative quantifiers typified by 
terms such as ‘most’, ‘at least half, or ‘as many as possible’. 
The membership function of relative quantifiers for a given 
pixel as given by the i' h classifier can be defined as (Herrera and 
Verdegay, 1996): 
Qpp, 
0 
РР;-о 
b-a 
1 
if PPì< a 
if a < pp i < b 
if pp, > b 
(1) 
Figure 2. Linguistic Quantifier at least half with the parameter 
pair (0, 0.5). 
4.5 Evaluation of the Proposed Method 
The overall classification accuracies of individual classifiers, 
based on the reference data, w'ere evaluated first and the overall 
accuracy of the best classifier served as a reference. Two of the 
most wddely used probability combination strategies were also 
tested and compared to the proposed method. These strategies 
include: Maximum Rule (MR); and Weighted Sum (WS). A 
detailed description of these combination methods can be found 
in Yager (1998). Since the overall accuracy is just a global 
measure for the performance of the combination process, two 
additional measures were used to evaluate the performance of 
the proposed combination method, namely: commission and 
omission errors. Unlike overall classification accuracy, 
commission and omission errors clearly show how the 
performance of the proposed method improves or deteriorates 
for each individual class in the combined classifiers. 
Commission errors are the percent of incorrectly identified 
pixels associated with a class, and omission errors are the 
percent of unrecognized pixels that should have identified as 
belonging to a particular class. All the methods proposed in this 
research w'ere implemented in Matlab (R2008b) environment. 
With parameters a,b E [0, 1], and pp, is the class membership 
of the pixel as obtained for the i‘ h classifier. Then, Yager (1988) 
proposed to compute the weights based on the linguistic 
quantifier represented as follows: 
Wpp ' =Qpp > (ir}~ Qpp ‘ ("hr) (2) 
Qm is the membership functions of relative quantifiers for the 
pixel as obtained for the i th classifier. J, is the order of the i' h 
classifier after ranking Q pp values of the pixel, for all 
classifiers, in a descending order. N is the total number of 
classifiers. 
The relative quantifier ‘at least half with the parameter pair (0, 
0.5) for the membership function Q PP in equation 1 as 
graphically depicted in figure 2 was used. Depending on a 
particular number of classifiers N, 3 in our case, and by using 
equation 2, the corresponding weighting vector of the given 
pixel, W PP = [w PPh , w’ppv] can be obtained. Finally, the 
probability based on FMV (P FMv ) can be calculated as follows: 
P Fm - = arg max 
k 
wdth k is the number of classes. 
5. RESULTS AND ANALYSIS 
5.1 Comparison with Existing Fusion Algorithms 
The overall classification accuracies of individual classifiers, 
based on the reference data, are given in Table 4. SVMs 
perform the best with 96.8% average overall classification 
accuracy, followed by SOM and CTs with average overall 
classification accuracies of 95.5% and 93.7% respectively. The 
overall accuracy of the best classifier served as a reference in 
the following. 
Classification 
(%) 
accuracy 
Test area 
SOM 
CT 
SVMs 
UNSW 
96.8 
95.05 
96.9 
Bathurst 
95 
92.85 
96.5 
Fairfield 
96.8 
96.15 
97 
Memmingen 
95 
90.75 
96.6 
Mean 
95.9 
93.7 
96.75 
SD 
1.04 
2.40 
0.24 
Table 4. Performance evaluation of single classifiers for the 
four test areas. 
The improvement in overall classification accuracies achieved 
by the combination method compared with the best individual 
classifier, SVMs, w'as determined as show’n in Figure 3. For the 
I 
vv pp 
pp , rrt 
(3)