## 3 Materials and Methods
### Feature Selection
#### 3.2.2 HOS Features
The \(f_{1}\) feature is given by \[f_{1}=\sum_{i=1}^{N_{g}}\sum_{j=1}^{N_{g}}(\bar{p}(i,j))^{2}.\] (7)2. The contrast feature, \(f_{2}\), is a measure of the status of the local variations in the intensity values between pixels of an image. The higher the local variation value, the larger the contrast. Contrast also represents a measure of intensity variation between a reference pixel and its neighbors in the image. Low contrast reflects low-intensity differences in \(\mathbf{\tilde{P}}\) and vice versa. The value of the contrast feature, \(f_{2}\), is given by \[f_{2}=\sum_{n=0}^{N_{S}-1}n^{2}\left(\sum_{i=1}^{N_{S}}\sum_{j=1}^{N_{S}} \tilde{p}(i,j)\right).\] (8) There is no local contrast for a cell by itself in the \(\mathbf{\tilde{P}}\) matrix, so the value of \(|i-j|\) plays the role of a weight. There is no contrast if \(|i-j|=0\). The contract keeps increasing as \(|i-j|\) becomes larger. 3. The correlation feature, \(f_{3}\), indicates that pairs of adjacent pixels are correlated (positively, neutrally, or negatively) in the retinal image. It measures the amount of linear dependence of the gray-level values in the image. Let \(\tilde{p}_{x}(i)=\sum_{j=1}^{N_{S}}\tilde{p}(i,j)\) be the \(i\)th element in the marginal probability matrix obtained by summing the columns of \(\mathbf{\tilde{P}}\) and \(\tilde{p}_{y}(j)=\sum_{i=1}^{N_{S}}\tilde{p}(i,j)\) be the \(j\)th element in the marginal probability matrix obtained by the summation of the rows of \(\mathbf{\tilde{P}}\). Additionally, let \(\mu_{x}=\sum_{i=1}^{N_{S}}i\tilde{p}_{x}(i)\) and \(\mu_{y}=\sum_{j=1}^{N_{S}}j\tilde{p}_{y}(j)\) be the means of \(\tilde{p}_{x}\) and \(\tilde{p}_{y}\). Further, let \[\sigma_{x}=\sqrt{\sum_{i=1}^{N_{S}}(i-\mu_{x})^{2}\tilde{p}_{x}( i)},\] \[\sigma_{y}=\sqrt{\sum_{j=1}^{N_{S}}(j-\mu_{y})^{2}\tilde{p}_{y}( j)},\] be the standard divisions of \(\tilde{p}_{x}\) and \(\tilde{p}_{y}\). Then the correlation feature, \(f_{3}\), is given by \[f_{3}=\sum_{i=1}^{N_{S}}\sum_{j=1}^{N_{S}}\frac{ij\ \tilde{p}(i,j)-\mu_{x}\mu_{y}}{\sigma_{x}\sigma_{y}}.\] (9)
4. The sum of squares, or variance, feature, \(f_{4}\), is a measure of the dispersion of the values around the mean and describes the higher weights that differ from the mean value. Let \(\mu\) be the mean of the \(\mathbf{\tilde{P}}\) matrix, then the variance features, \(f_{4}\), are given by \[f_{4}=\sum_{i=1}^{N_{S}}\sum_{j=1}^{N_{S}}(i-\mu)^{2}\ \tilde{p}(i,j).\] (10)
5. The inverse difference moment feature, \(f_{5}\), shows how the distribution is close to the elements of the diagonal of \(\mathbf{\tilde{P}}\). Conceptually, as the inverse difference moment decreases, the contrast increases. This feature is given by \[f_{5}=\sum_{i=1}^{N_{S}}\sum_{j=1}^{N_{S}}\ \frac{1}{1+(i-j)^{2}}\ \tilde{p}(i,j).\] (11)The sum average, \(f_{6}\), feature is the higher weight to the higher index of the marginal \(\bar{\mathbf{P}}\) matrix. If \(\bar{p}_{x+y}(i)\) is the probability of the \(\bar{\mathbf{P}}\) matrix, then the \(f_{6}\) feature is given by  
\[f_{6}=\sum_{i=2}^{2N_{g}}i\bar{p}_{x+y}(i), \tag{12}\]  
where  
\[\bar{p}_{x+y}(k)=\sum_{ \begin{subarray}{c}i=1\\ i+j=k\end{subarray} }^{N_{g}}\sum_{j=1}^{N_{g}}\bar{p}(i,j),\ \ \ k=2,3,\ldots,2N_{g}.\]
7. The sum variance feature, \(f_{7}\), is a measure of the higher weights that differ from the sum entropy value (the \(f_{8}\) feature below) of the marginal \(\bar{\mathbf{P}}\) matrix, and is given by \[f_{7}=\sum_{i=2}^{2N_{g}}(i-f_{8})^{2}\,\bar{p}_{x+y}(i).\] (13)
8. The sum entropy feature, \(f_{8}\), is calculated as \[f_{8}=-\sum_{i=2}^{2N_{g}}\bar{p}_{x+y}(i)\log\bar{p}_{x+y}(i).\] (14) Since \(\log(0)\) is undefined, whenever a \(0\) probability is encountered during the computations, it should be replaced by an arbitrarily small positive value. 9. The entropy feature, \(f_{9}\), is a concept of information theory, estimating the randomness of pixel intensities in the image. As such, its value is zero for a constant image. It is given by \[f_{9}=-\sum_{i=1}^{N_{g}}\sum_{j=1}^{N_{g}}\bar{p}(i,j)\log\bar{p}(i,j).\] (15)
10.