International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
r 
hi ————— (1) 
  
— controls the amount of fuzziness. 
number of classes of interest. 
Mahalanobis distance of the pixel / from the 
where r 
€ 
Il 
Il 
dy 
mean of the class &. It is computed by: 
dj, -(I-m, Y V,(I-m,) (2) 
m, = mean vector of the class À. 
Vi = inverse of the covariance matrix of the class À. 
This model involves that for each pixel the sum of the fuzzy 
memberships in all classes is equal to one. 
> hy =17/ (3) 
k=] 
2.2. Change detection methods 
2.2.1. Comparative analysis-based change detector 
In this approach, the fuzzy classifier presented above is used to 
produce independent classifications for two images. 
Traditionally, (i.e. with hard classifiers) the change is detected if 
the labels of a given pixel in dates /, and #, are different. 
However, using fuzzy classifiers we do not have single class 
labels to compare. Instead, we have the degree of membership 
of each pixel in each of the classes of interest. In such a case 
arithmetic operators as well as ranking techniques do not lead to 
a result which can be considered as a membership value, despite 
of being real numbers comprised between 0 and 1. 
Consequently, we use triangular norms to define change and no 
change classes (Deer, 1998). 
Hence, the fuzzy class membership of a pixel x in the class 
A at t, is described by 71, , (t ). Similarly, the membership in the 
class B at the date 1, is described by hp (6, ). To inspect at 
what point this situation is truth, we evaluate the fuzzy 
membership in the change class (A, B), that is defined as 
Min(h., ti). ha (ts )) (4) 
2.2.2. Simultaneous analysis-based change detector 
This change detector considers the bitemporal space as a single 
date space. Thus, classes of interest are either change or no 
change classes. The situation in which the pixel x was in class 4 
at /, and is in class B at f; is described by hs atis t2) 
according to (1). In this case, covariance matrices and mean 
710 
vectors of the Mahalanobis distance are computed using all 
spectral bands of the two images. 
2.3. Combination scheme 
The concept of the combination scheme is to pool decisions or 
classification scores from multiple information sources into a 
single composite score by applying a fuzzy integral with respect 
to a designated fuzzy measure. Thus, in this paper, we combine 
outputs of simultaneous analysis ($4) and comparative analysis 
(CA) based change detectors as shown in figure 1. 
  
    
    
    
        
  
  
    
  
Classifier 
Classifier 
Combiner 
   
EU 7 Classifier 
| 
| 
1 
| 1 
I ‘he 2! 
| i Change 
! ! J detection map 
I I ; 
1 
) SA 
1 DEE i ! 
I 
| 
1 . . ~ 
ie ee me mee) » Multispectral images of two dates 
Figure 1. Description of change detector combination 
The fuzzy integral combines objective evidences given in the 
form of fuzzy grade memberships, with a subjective evaluation 
of the reliability of individual change detectors. The concept of 
the fuzzy integral and the associated fuzzy measure was 
originally introduced by Sugeno in the early 1970s in order to 
extend the classical (probability) measure through relaxation of 
the additivity property (Cho and Kim, 1995). A formal 
definition of the fuzzy measure is as follows. 
Fuzzy measure: let Z be a finite set of elements. A set function 
Z : 
gua > [0, 1] with: 
1. glg)=0 
(Z)=1 
3. cg A)< g(B) if Ac B 
is called fuzzy measure. This measure does not follow the 
addition rule. In other words, for two sets 4, B c Z and 
ve 
2. 
ve 
satisfying AM B = $ , the equation (5) does not apply. 
g(AUB)= g(4)+2(8) (5) 
To overcome this limitation, Sugeno introduced the so 
called g; fuzzy measure. 
  
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