## 2 Materials and Methods
### Similarity Functions  
Similarity functions [13], also called indexes or indicators based on qualitative (binary) attribute data, were first used in ecology for grouping of either biotical communities or ecologically related species. The literature refers to over 50 similarity or dissimilarity indexes. The following indexes are derived from Definition 8.  
**Definition 9**.: _Similarity function_  
_A similarity function or indicator \(\Psi(\alpha,\beta)\) is a measure of the degree of similarity between two vectors \(\alpha,\beta\) of the \(u\)-dimensional binary space. \(\Psi(\alpha,\beta)\) has the following properties:_  
\[\Psi(\alpha,\beta)=\left\{\begin{array}{c}0\leq\Psi(\alpha,\beta)<1\ \mbox{if}\ \alpha\
eq\beta\\ 1\ \mbox{if}\ \alpha=\beta\end{array}\right.\]  
\[\Psi(\alpha,\beta)=\Psi(\beta,\alpha).\]  
The operation \(\triangle\) is used to define different similarity functions between \(\alpha,\beta\) such as:  
1. Co-linearity indicator The co-linearity indicator is proposed based on the \(\mathcal{R}(\alpha,\beta)=[\alpha_{i}=\beta_{i}]\) relation with matches \(<\alpha,\beta>\), or true positive relations. \[\Psi_{ICo}(\alpha,\beta)=\lambda=\frac{<\alpha,\beta>}{\sqrt{\mathcal{N}( \alpha)}\sqrt{\mathcal{N}(\beta)}}\] (2) If \(\lambda=1\), then \(\alpha,\beta\) are co-linear. If \(\lambda=0\), then \(\alpha,\beta\) are orthogonal.
2. Maximum sensitivity area indicator This indicator relates the true positive rate (TPR) and the false positive rate (FPR). The relation between the TPR and the FPR is represented by the well-known receiver operating characteristic (ROC) curve [14]. If the TPR tends to 1 and the FPR tends to 0, then the correlation between \(\alpha,\beta\) is high. The following relations define the metrics of the indicator: \[TPR=\frac{1}{1+k_{1}}\ \mbox{FPR}=\frac{1}{1+k_{2}}\ k_{1}=\frac{<\alpha \,\overline{\beta}>}{<\alpha,\beta>}\ k_{2}=\frac{<\overline{\alpha},\overline{ \beta}>}{<\overline{\alpha},\beta>}\] \[\Psi_{IS}(\alpha,\beta)=\Psi_{area-max}(\alpha,\beta)\ =0.5\bigg{(}1+\frac{1}{1+k_{1}}-\frac{1}{1+k_{2}} \bigg{)}\] (3) If  
If \(\Psi_{IS}(\alpha,\beta)\) approaches 1 (maximum area), then the results obtained in the classification is accurate (Figure 2A).
3. Minimum distance indicator This indicator measures the minimum distance between the points (TPR, FPR) and (0,1) in the ROC space, as shown in Figure 2B.  
\[\Psi_{dmin}(\alpha,\beta)=\sqrt{\left(1-\left(\frac{1}{1+k_{1}}\right)\right)^ {2}+\left(\frac{1}{1+k_{2}}\right)^{2}} \tag{4}\]  
If \(k_{1}\) approaches 0 and \(k_{2}\) approaches infinite, then \(\Psi_{dmin}(\alpha,\beta)\) tends to 0 (minimum distance). The conditions for \(k_{1}\) and \(k_{2}\) are identical to those of the previous indicator.
4. Coverage indicator of the segmented area The coverage or superposition indicator compares the reference set \(\alpha\) with the segmented one \(\beta\), and presents the one-to-one correspondence between the sets pixels. The coverage indicator of segmented area \(\Psi_{IC}(\alpha,\beta)\) is defined as [13]:  
\[\Psi_{IC}(\alpha,\beta)=\frac{2{<\alpha,\beta>}}{\|\alpha\|+\|\beta\|} \tag{5}\]  
where \(0\leq\Psi(\alpha,\beta)<1\) if \(\alpha\
eq\beta\) and 1 if \(\alpha=\beta\).