# Retinal Vessel Segmentation Using Math-Inspired Metaheuristic Algorithms
## 3 Results
### Statistical Analysis  
The results produced by the metaheuristic algorithms were tested and verified in terms of the Se, Sp, Acc and standard deviation metrics due to their non-deterministic nature. Furthermore, the Wilcoxon rank-sum test was applied to statistically validate the obtained numerical results and investigate the significant differences among the algorithms [56]. The Se, Sp, and Acc metrics can be defined according to the following expressions:  
\[Se=\frac{TP}{(TP+FN)} \tag{34}\]  
\[Sp=\frac{TN}{(TN+FP)} \tag{35}\]\[Acc=\frac{(TP+TN)}{(TP+FN+TN+FP)} \tag{36}\]  
where true positives (TP) are the correctly classified vessel pixels, false negatives (FN) are the incorrectly classified vessel pixels, true negatives (TN) are the correctly classified background pixels, and, finally, false positives (FP) are the incorrectly classified background pixels [55]. Performance measures and statistical analyses were applied separately to all 40 images in the DRIVE and STARE databases in order to perform a more comprehensive and global analysis. The performances of the algorithms in terms of Se, Sp, and Acc on the DRIVE and STARE databases are given in Tables 2 and 3, respectively, where higher Se, Sp, and Acc values correspond to better segmentation performance. From the tables, it can be seen that all of the algorithms converged to similar values in terms of Se, Sp, and Acc for almost all retinal images in the DRIVE and STARE databases. Figure 5: Results after applying CSA-, TSA-, AOA-, CMDO-, GOBC-PA-, and SCA-based segmentation to the images in Figure 1a,b. Figure 6: Results after applying CSA-, TSA-, AOA-, GNDO-, GOBC-PA-, and SCA-based segmentation to the images in Figure 2a,b. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c c} \hline \multicolumn{11}{c}{**Sensitivity**} & \multicolumn{1}{c}{**Sensitivity**} & \multicolumn{1}{c}{**Sensitivity**} & \multicolumn{1}{c}{**Sensitivity**} & \multicolumn{1}{c}{**Accuracy**} & \multicolumn{1}{c}{**Sensitivity**} \\ \hline
**Image** & **CSA** & **TSA** & **AAOA** & **CNDO** & **GORC-PA** & **SCA** & **CSA** & **TSA** & **  
When the mean values obtained for the DRIVE images are examined, the performance of the TSA in terms of correctly classified TP pixels seems better than that of the other algorithms due to its highest mean Se value. On the other hand, the highest mean Sp value produced by the GOBC-PA algorithm demonstrates that it is able to produce better performance in terms of correctly classified TN pixels when compared to other algorithms. Finally, the highest mean Acc value obtained for the TSA proves its superior performance in terms of the total number of image pixels that are correctly classified. From the mean values obtained for the STARE images, it can be seen that the CSA produced the highest performance in terms of the mean Se value, which corresponds to the correctly classified TP pixels. Moreover, the highest mean Sp value obtained for the SCA shows that it produced the best results in terms of correctly classified TN pixels. Finally, the performance of the TSA in terms of correctly classified total image pixels seems better than that of the other algorithms due to its highest mean Acc value. Another important statistical performance metric for metaheuristic algorithms is the standard deviation, which indicates the ability of an algorithm to reach similar results in each run. In particular, smaller standard deviation values prove the stability and robustness of an algorithm. The standard deviation values of each algorithm obtained for 20 random runs are shown in Figure 7. It can be seen that the GOBC-PA algorithm reached the lowest standard deviation values on both the DRIVE and STARE images. It can be concluded that the GOBC-PA algorithm exhibits somewhat stabler and more robust behavior when compared to other algorithms, although the results were generally similar. It can also be seen that the performance of the TSA and AOA seems to be slightly worse than that of the other algorithms in terms of the standard deviation.