## 3 Methodology
### Fuzzy Image Classification  
In traditional classification methods such as minimum distance method, each pixel or each segment in the image will have an attribute equal to 1 or 0 expressing whether the pixel or segment belongs to a certain class or not, respectively. In fuzzy classification, instead of a binary decision-making, the possibility of each pixel/segment belonging to a specific class is considered, which is defined using  
Figure 1: Flow chart of the presented method.  
membership functions. A membership function offers membership degree values ranging from 0 to 1, where 1 means fully belonging to the class and 0 means not belonging to the class [24].  
Implementing fuzzy logic ensures that the borders are not crisp thresholds any more, but membership functions within which each parameter value will have a specific probability to be assigned to a specific class are used. Appending more parameters to this classification, for example, using NIR ratio and NDVI for vegetation classification, better results will be achieved. Using fuzzy logic, classification accuracy is less sensitive to the thresholds.  
\(\mu_{A}\) is a fuzzy membership function over domain \(X\). \(\mu_{A}(x)\) is called membership degree, which ranges from 0 to 1 over domain \(X\)[23, 24]. \(\mu_{A}(x)\) can be a Gaussian, Triangular, Trapezoidal, or other standard functions depending on the application. In this research, trapezoidal and triangular functions are used (Figure 2). The associated formulas are given in Equations (1) and (2) [25, 26].  
Triangular function:  
\[\mu_{A}(x)=1-\frac{|a-x|}{\lambda},\text{for }0\leq|\text{a}-x|\leq\lambda; \mu(x)=0 \tag{1}\]  
Trapezoidal function:  
\[\mu_{A}(x)=\min\left\{2-2\left(\frac{|a-x|}{\lambda}\right),1\right\},\text{ for a}-\lambda\leq|\text{a}-x|\leq\text{a}+\lambda\,;\mu(x)=0 \tag{2}\]  
where, \(a\) is the \(x\) coordinate of the middle point of the trapezoidal function or the x coordinate of the peak of the triangular function and \(\lambda\) equals half of base of triangle or half of the long base of the trapezoidal. All the parameters of the functions are specified based on human expertise [25].  
In fuzzy rule-based systems, lingual variables are introduced, which replace the crisp thresholds. For example, instead of defining vegetation with a threshold for NDVI, a lingual value, such as high, medium, or low, with a specific fuzzy function is identified which assigns a membership degree to  
Figure 3: Lingual variable example.  
Figure 2: Typical (**a**) Trapezoidal. (**b**) Triangular fuzzy functions which are used in this study.  
specific ranges of NDVI. Figure 3 shows an example of lingual variables [25; 26; 27]. In this research, the only lingual variables which are useful for this study are defined.  
The other important specification for a fuzzy rule-based system is a fuzzy inference system, which uses Fuzzy rules for decision making. In this project the inference system of eCognition software is used. More details of fuzzy classification in eCognition software are given in [28].  
In the presented method, the specifications of each object are tested using the fuzzy rules defined for each class based on the hierarchy mentioned in Section 3. Each segment receives a degree specifying the similarity of the segment to each class. The segments which have high similarity degrees will be assigned to the associated class after deffuzzification of the results. This step is also done in eCognition software.  
In the following, the details of the parameters used for different object classification are presented.