# [PERSON]'s Entropy Based Multilevel Thresholding Using a Novel Meta-Heuristics Algorithm
## 4 Simulation Results and Discussion
### Quantitative Evaluation of Segmented Results
However, the HRO-based threshold segmentation algorithm has achieved the maximum fitness value in these two cases, indicating that the algorithm is more effective than other algorithms. The accuracy of HRO-Renyi to find the optimal solution is the highest among all methods, but the number of iterations required to find the optimal solution is more than that of PSO-Renyi and ALO-Renyi, it can converge at about 40 to 50 iterations. When the number of iterations is 50, the optimal solution obtained by HRO-Renyi is already better than other methods, DE-Renyi and SSA-Renyi need more iterations to converge, they probably need about 70 iterations, while the number of iterations required for convergence of WOA-Renyi and GA-Renyi is close to HRO-Renyi, but the accuracy is not as good as HRO-Renyi. In this section, STD is used to measure the stability of each optimization algorithm in each threshold segmentation algorithm. STD is the arithmetic ordinary root of variance, and the average of the distance (deviation from average) of each data deviation. STD can reflect the dispersion degree of a data set. Table 3 represents the STD of each optimization algorithm after 50 tests in Renyi's entropy (Bold is best). When the STD is smaller, the algorithm is more stable. After the minimum STD of all methods is calculated, HRO-Renyi has the lowest STD in 28 of 32 experiments, accounting for 87.5% of all experiments, the minimum STD times of other methods are: ALO-Renyi has 12 times, DE-Renyi has 8 times, SSA-Renyi has 6 times, WOA-Renyi has 2 times, PSO-Renyi has 9 times, and GA-Renyi has 0 times. When the threshold-level is 2, except for WOA-Renyi, SSA-Renyi, and GA-Renyi, other methods have a STD of 0 or close to 0 in most test images, which means that the optimal solutions obtained by these methods in 50 experiments are all equal and have strong stability. As the number of thresholds increases, the STD increases, the optimization algorithm easily falls into a local optimum, which leads to the instability of the algorithm. As could be seen from the Table 3, when the threshold-levels k = {2, 4}, STD of HRO-Renyi method is mostly 0 or close to 0. When the threshold-levels k = {6, 8}, the maximum STD is 0.176. Compared with other algorithms, HRO has the minimum STD and the best stability. Figure 5: Convergence curve of the maximum fitness functions of each algorithm in Renyi’s entropy. It has been manifested above that the HRO-Renyi method has achieved good results in terms of the best fitness function and algorithm stability. HRO-OTSU is also used to segment the experimental image, quantitative results are defined by PSNR, SSIM. To improve the richness of experimental results, the experimental image is divided into two parts in this discussion, one is the image in the standard data set, and the other is the SEM image of cement. The optimal PSNR and SSIM values obtained by HRO-OTSU and HRO-Renyi are given in Table 4. It could be found that, in terms of PNSR, performance of HRO-Renyi on the standard data set is basically the same as that of HRO-OTSU. However, on the SEM image of cement data set, no matter if threshold-levels k = {2, 4, 6, 8}. HRO-Renyi is larger than HRO-OTSU, which shows that HRO-Renyi has good robustness and good segmentation quality. In terms of SSIM, HRO-Renyi performs slightly better than HRO-OTSU on the standard data set, but when threshold-levels k = {6, 8}, the former is often larger than the latter, and on the SEM image of cement data set, regardless of whether threshold-levels k = {6, 8}, HRO-Renyi is larger than HRO-OTSU, which shows that performance of HRO-Renyi in multi-dimensional threshold segmentation is better than HRO-OTSU. Therefore, the HRO-Renyi method is more robust and more suitable for multi-dimensional threshold segmentation of SEM images of cement.