# Retinal Vessel Segmentation Using Math-Inspired Metaheuristic Algorithms
## 3 Results
### Convergence Analysis  
Convergence speed is one of the most important metrics reflecting the performance of metaheuristic algorithms, indicating the number of cycles at which the algorithm reaches the global solution. Figure 8 demonstrates the convergence speeds obtained in terms of the mean MSE values for 20 random runs. As can be seen from the figure, the performances of the TSA and AOA in terms of convergence speed and mean squared error seem slightly worse than those of the other algorithms. On both databases, the TSA reached its global optimum at approximately cycle 40, while the AOA reaches it at approximately cycle 95. In contrast, the other algorithms reached similar MSE values at close convergence rates on both the DRIVE and STARE images. In addition to the convergence speed, the CPU time required for each algorithm to reach the global solution is another important parameter. Analyses were carried out using a computer with a 3.4 GHz Intel i7-6700 CPU, 16 GB RAM, and 64 bit Windows 10 Pro. The minimum MSE values reached by the algorithms and the related CPU times are given in Table 5 for both databases. Evaluating Figure 8 and Table 5 simultaneously, it can be seen that, although they presented a slower rate of convergence, the TSA and AOA produced better results in terms of the CPU time for both databases. Meanwhile, the GNDO and GOBC-PA algorithms produced worse results in terms of the CPU time despite their higher convergence rates. Due to the structural differences between the metaheuristic algorithms, the computational complexity of each algorithm was also analyzed in detail. In particular, in order to compare their computational complexities, the algorithms were analyzed in terms of the _Number of Function Evaluations (NFEs)_ metric, which indicates the number of times the objective function is evaluated during the optimization process. In addition to the _NFEs_ for each algorithm, the _Elapsed CPU time for NFEs_ and the _Percentage of this time in the total CPU time_ were also calculated. The results obtained using the _NFEs_ analyses are given in Table 6. As can be seen from the table, the relationship between the NFEs and CPU time values can be defined as linear. However, by analyzing Figure 8 and Table 6 simultaneously, it can be \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Algorithms** & **Better Than** \\ \hline \multirow{5}{*}{\
\begin{tabular}{c} **DRIVE** \\ (Figure 1b) \\ \end{tabular} } & TSA & - \\ \cline{2-3}  & AOA & TSA (\(3.50\times 10^{-3}\)) \\ \cline{2-3}  & SCA & TSA (\(2.4621\times 10^{-4}\)) \\ \cline{2-3}  & CSA & TSA (\(2.2337\times 10^{-5}\)) & AOA (\(4.1790\times 10^{-4}\)) & SCA (\(2.2337\times 10^{-5}\)) \\ \cline{2-3}  & GNDO & TSA (\(1.8307\times 10^{-5}\)) & AOA (\(3.6315\times 10^{-4}\)) & SCA (\(1.8307\times 10^{-5}\)) \\ \cline{2-3}  & GOBC-PA & TSA (\(1.0269\times 10^{-5}\)) & AOA (\(2.4155\times 10^{-4}\)) & SCA (\(1.0269\times 10^{-5}\)) \\ \hline \multirow{5}{*}{
\begin{tabular}{c} **STARE** \\ (Figure 2a) \\ \end{tabular} } & TSA & - \\ \cline{2-3}  & AOA & - \\ \cline{2-3}  & SCA & TSA (\(1.2335\times 10^{-4}\)) AOA (\(1.70\times 10^{-3}\)) \\ \cline{2-3}  & CSA & TSA (\(2.2337\times 10^{-5}\)) & AOA (\(4.1790\times 10^{-4}\)) & SCA (\(2.2337\times 10^{-5}\)) \\ \cline{2-3}  & GNDO & TSA (\(1.4191\times 10^{-5}\)) & AOA (\(3.0345\times 10^{-4}\)) & SCA (\(1.4191\times 10^{-5}\)) \\ \cline{2-3}  & GOBC-PA & TSA (\(1.0269\times 10^{-5}\)) & AOA (\(2.4155\times 10^{-4}\)) & SCA (\(1.0269\times 10^{-5}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Wilcoxon rank-sum test results obtained for the algorithms (\(p<0.05\)). concluded that the _NFEs_ value does not cause an important variation in the minimum MSE values reached. The analysis results can be summarized as follows: for both the DRIVE and STARE databases, all of the algorithms produced similar results in terms of correctly classified and misclassified pixels. The CSA stands out, with its simple algorithm structure, when compared to the other algorithms.