## 4 Experimental Results and Analysis
### Comparative Experiments and Analysis
#### 4.5.2 Comparison with State-of-the-Art Methods  
The performance of Cauchy-TB was evaluated against nine recently proposed methods using various popular optimization algorithms that were selected as standards for comparison in this domain, namely, HHO (Harris Hawks Optimization) [63], LSHADE (Large-Scale Differential Evolution) [64], DE (Differential Evolution) [50], PSO (Particle Swarm Optimization) [65], GWO (Gray Wolf Optimizer) [66], SCA (Sine-Cosine Algorithm) [67], WOA(Whale Optimization Algorithm) [68], ALO (Ant Lion Optimizer) [69], and CSA (Cuckoo Search Algorithm) [70]. All experiments were conducted on a small subset consisting of eight \(Z\)-plane-cut brain slices (Z1, Z2, Z5, Z10, Z36, Z72, Z108, and Z144) in BrainWeb. Table 6 shows the PSNR, SSIM, and FSIM metrics of Cauchy-TB applied to weight images using one, two, and three thresholds for segmentation. Our method achieved excellent results on all three metrics for the eight images, with the highest PSNR reaching 31.63 and the highest FSIM reaching 0.992. Typically, a PSNR value of 30 is considered a good quality level, and an FSIM value of 1 is considered completely similar. Although the SSIM value was slightly smaller, the results were still within an acceptable range. The results of a comparative experiment with these 10 algorithms are presented in Table 7. The values obtained with the proposed Cauchy-TB are shown in the last two rows of the table. Considering that all nine standard algorithms have stochasticity, the results in Table 7 reveal a consistent trend across different threshold numbers and all optimization algorithms: the performance of Cauchy-TB surpassed the other algorithms on the metrics of PSNR and FSIM, while its slight disadvantage in SSIM stayed within acceptable limits. For eight brain MRI images, Table 8 demonstrates the segmentation results for two threshold selections (two and three). The corresponding intensity histograms for the original images and two results are shown below, with the threshold positions marked on the histogram. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**Group**} & \multicolumn{3}{c}{**Tsallis-BE**} & \multicolumn{3}{c}{**Cauchy-TB**} \\ \cline{2-7}  & **Accuracy** & **JC** & **DSC** & **Accuracy** & **JC** & **DSC** \\ \hline BrainWeb-1 & **0.9666** & **0.9399** & **0.9690** & 0.9653 & 0.9378 & 0.9679 \\ BrainWeb-2 & 0.9354 & 0.8554 & 0.9210 & **0.9526** & **0.8862** & **0.9392** \\ BrainWeb-3 & 0.9376 & 0.8824 & 0.9372 & **0.9482** & **0.9001** & **0.9472** \\ BrainWeb-4 & 0.9716 & 0.9624 & 0.9808 & **0.9754** & **0.9675** & **0.9835** \\ BrainWeb-5 & 0.9768 & 0.9686 & 0.9840 & **0.9773** & **0.9695** & **0.9845** \\ BrainWeb-6 & 0.9775 & **0.9699** & **0.9847** & **0.9776** & 0.9698 & **0.9847** \\ BrainWeb-7 & 0.9809 & 0.9754 & 0.9875 & **0.9835** & **0.9790** & **0.9894** \\ BrainWeb-8 & 0.9737 & 0.9680 & 0.9837 & **0.9816** & **0.9778** & **0.9888** \\ BrainWeb-9 & 0.9709 & 0.9609 & 0.9800 & **0.9770** & **0.9693** & **0.9844** \\ BW-Average & 0.9657 & 0.9425 & 0.9698 & **0.9709** & **0.9508** & **0.9744** \\ \hline \hline \end{tabular} The accuracy, JC, and DSC were computed for sliced images from the BrainWeb datasets using two methods for ternary segmentation. The best result is indicated by the bold number.  
\end{table} 
Table 5: Results of comparative experiments for ternary segmentation.