# Retinal Vessel Segmentation Using Math-Inspired Metaheuristic Algorithms
## 4 Discussion  
In this work, math-inspired metaheuristic algorithms--including the CSA, TSA, AOA, GNDO, GOBC-PA, and SCA--were implemented as clustering approaches for high-accuracy retinal vessel segmentation. Analyses were carried out on both healthy and diseased retinal images from the DRIVE and STARE databases. From the obtained segmentation results, it can be seen that each of the algorithms successfully distinguished the vessel and background pixels of the retinal images. The close and high Se, Sp, and Acc values produced by the algorithms prove the robustness of each algorithm in terms of their statistical performance. In addition, due to the lower standard deviation values of the CSA, GNDO, and GOBC-PA algorithms, it can be concluded that these algorithms are slightly more stable in the context of clustering-based retinal vessel segmentation. Furthermore, according to the results of the Wilcoxon rank-sum test carried out among the algorithms, it can be expressed that the CSA, GNDO, and GOBC-PA algorithms produced statistically better results compared to the TSA, AOA, and SCA. Finally, when the convergence performances of the algorithms were evaluated as a whole, it was found that, despite their higher convergence rates, the GNDO and GOBC-PA algorithms present slightly worse performance in terms of CPU time. It can additionally be expressed that the TSA presented the lowest NFEs rates in both databases compared to the other algorithms. It can consequently be expressed that the considered math-inspired metaheuristic algorithms can successfully be used in the context of biomedical image processing.  
When the efficiency of math-inspired metaheuristic algorithms is compared to that of evolutionary-based, swarm-intelligence-based, and physics-based metaheuristic algorithms [52; 53], it can be seen that these algorithms are able to produce similar results in terms of retinal vessel segmentation. The MSE, CPU time, and NFEs values obtained using the considered math-inspired metaheuristic algorithms reveal that these algorithms perform well in terms of convergence speed. On the other hand, the Se, Sp, Acc, standard deviation, and Wilcoxon rank-sum test results obtained demonstrate the good statistical performance and robustness of these algorithms. Consequently, it can be expressed that math-inspired metaheuristic algorithms are capable of producing similar performances to other algorithms in the considered context, despite their simpler algorithm structures and lower number of control parameters.  
Conceptualization, M.B.C.; methodology, M.B.C. and S.A.; software, M.B.C.; validation, M.B.C. and S.A.; formal analysis, M.B.C. and S.A.; investigation, M.B.C. and S.A.; writing--original draft preparation, M.B.C.; writing--review and editing, M.B.C. and S.A. All authors have read and agreed to the published version of the manuscript.  
The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.  
Not applicable.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Algorithms**} & \multirow{2}{*}{**Total NFEs**} & **DRIVE** & **STARE** \\  & & **(Figure 1b)** & **(Figure 2a)** \\ \hline
**GOBC-PA** & & 5.0150 s & 6.5470 s \\  & & (78.09\% of the total CPU time) & (78.16\% of the total CPU time) \\ \hline
**SCA** & 1010 & 2.0750 s & 2.6120 s \\  & & (76.91\% of the total CPU time) & (77.20\% of the total CPU time) \\ \hline \hline \end{tabular}
\end{table}
Table 6: _Cont._Data Availability Statement: Data are contained within the article. Conflicts of Interest: The authors have no relevant financial or non-financial interests to disclose. The authors declare that they have no conflict of interests.