## 3 Methodology
### Fuzzy Image Classification
#### 3.2.3 Road  
Roads are elongated objects with smooth surfaces having low variations in the gray values. Therefore, criteria which show lengthened objects and gray value variances can be used for defining roads. Classifying road segments without any ancillary data is a challenging issue. In this project, first strict limitations are used for road detection. Therefore, the real road segments will be classified as Road class. Then, using spectral similarity with neighbor objects, other parts of the road can also be correctly classified. In this study, 3 parameters called \(\text{l}_{\text{cm}}\),\(\text{l}_{\text{e}}\), and standard deviation are used for road detection which are explained in this section.  
\(\text{l}_{\text{cm}}\) **and \(\text{l}_{\text{e}}\):** Equation (9) and (10) are used for defining elongated objects [6].  
Figure 5: NDVI or NIR Ratio fuzzy function. In this study for NDVI, \(T_{1}=0.05\), \(T_{2}=\)0.25, For NIR Ratio: \(T_{1}=0.15\), \(T_{2}=0.4\).  
\[I_{cm}=\frac{2\sqrt{\pi.Area(object)}}{perimeter(object)} \tag{9}\] \[I_{e}=\frac{Area(object)}{[length(object)]^{2}} \tag{10}\]  
Having compared different class segments, one comes to the understanding that road segments have low value for \(I_{cm}\) and \(I_{e}\), while buildings have high values of them. The exact range for \(I_{cm}\) and \(I_{e}\) associated with road segments depends on the image spatial resolution, road width, _etc._, However, a rough estimation for the used GeoEye image is that \(I_{cm}\) for road segments is less than 0.5 and \(I_{e}\) is less than 0.1 (These numbers are slightly different in the used QuickBird imagery). Although there are some other segments belonging to roads which have higher values of \(I_{cm}\) and \(I_{e}\), this tight limitation decreases the risk of error of commission, which means it prevents non-road elongated segments to be miss-classified in road class. Later, as explained above, using contextual information, those unclassified road segments will be properly classified to road class (see Section 3.2.5). The fuzzy function defined for \(I_{cm}\) and \(I_{e}\) is presented in Figure 6.  
**Standard deviation (Std)**: Std shows variation of pixel values in a specific segment [28]. Since the surfaces of the roads are smooth with low variations in gray levels, the lower the standard deviation of segments, the higher the possibility to be assigned to the road class. The upper limit of standard deviation for roads depends on the radiometric resolution of the image and the distribution of gray values in the image histogram. For example in this project, \(T=150\) for standard deviation of GeoEye image (Figure 6).  
**Figure 6.** Fuzzy function for \(I_{cm}\), \(I_{e}\) or Standard deviation, which has the same shape, but with different T values.  
Equation (11) presents the associated fuzzy set for \(I_{cm}\), \(I_{e}\) or Standard deviation.  
\[\mu_{I}(x)=\begin{cases}1-\dfrac{x}{T},&\text{for }0\leq x\leq T\\ 0,&\text{otherwise}\end{cases} \tag{11}\]  
Considering the above specifications for road, the produced fuzzy rule is:  
If \(\mathbf{I_{cm}}\) is _low_ and \(\mathbf{I_{e}}\) is _low_ and \(\mathbf{Std}\) is _low_ then segment is \(\mathbf{Road}\).  
Using this rule, some narrow elongated features not belonging to road class might also be miss-classified as road. Therefore, in this study after applying this rule, all the road segments so produced are merged. Then, those narrow features are removed using a width threshold. Considering that the minimum width of a road is 3 m and spatial resolution of the image is 0.5 m, all road segments which have width less than 6 pixels are unlikely to belong to roads and they are omitted from it. This procedure improves error of commission.