# Improving Prostate Image Segmentation Based on Equilibrium Optimizer and Cross-Entropy
## 5 Experiments
### Statistical Analysis of Standard Test Images  
Within stochastic optimization algorithms (SOAs), it is possible that two different approaches will produce the same results. If this is true, we could say that the two methods are not statistically different. To determine if this happens and to what extent, we use the Kruskal-Wallis test, a non-parametric method that discerns if the median of two or more distributions is statistically different [51]. We could say that Kruskal-Wallis is the non-parametric counterpart of the one-way analysis of variance (ANOVA) [52]. In this article, the analysis is applied to the prostate MRIs, where each experiment considers thirty independent runs of the same configuration. Each configuration includes an algorithm and a specific number of thresholds. All statistical significance tests require the definition of a significance level to accept or reject the null hypothesis, which states that all data are coming from the same distribution. The test outputs a p-value, and if it is lower than the null hypothesis, we can be sure that the algorithms are significantly different. In Table 4, the column named Ranking indicates the rank of each method in terms of fitness; the left-most algorithm is considered the best, while the right-most is the worst. In the notation, we can observe horizontal lines originating from two or more algorithms; this indicates that such algorithms are not significantly different. Figure 3: Cameraman image segmented with 2, 3, 4, 5, 8, and 16 thresholds. This figure displays the segmentation results of the cameraman image using different numbers of thresholds, demonstrating the progressive refinement of image regions as the number of thresholds increases. Figure 4: Eleven transaxial-cut prostate MRI images. This figure presents a set of eleven transaxial-cut magnetic resonance (MR) images of the prostate. These images serve as the input dataset for evaluating the segmentation performance of the proposed algorithm. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline  & \multicolumn{4}{c}{**SC**} & \multicolumn{2}{c}{**PSO**} & \multicolumn{2}{c}{**FO**} & \multicolumn{2}{c}{**DE**} & \multicolumn{2}{c}{**CA**} & \multicolumn{2}{c}{**HS**} & \multicolumn{2}{c}{**Ranking**} \\ \hline
**Image** & **at** & **Mean** & **Sid** & **Mean** & **Sid** & **Mean** & **Sid** & **Mean** & **Sid** & **Mean** & **Sid** & **Mean** & **Sid** & **Mean** & **Sid** & **Mean** & **Sid** \\ \hline
**1** & 3 & 0.8640 & 0.6099 & 0.767 & 0.0012 & 0.9549 & 0.7722 & 0.7688 & 0.0000 & 1.0710 & 0.0073 & 0.7684 & 0.0012 & 0.8745 & 0.1412 & \multicolumn{1}{c}{_E0 GA  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline  & \multicolumn{4}{c}{**SO**} & \multicolumn{4}{c}{**SO**} & \multicolumn{4}{c}{**SO**} & \multicolumn{4}{c}{**IO**} & \multicolumn{4}{c}{**Dr**} & \multicolumn{4}{c}{**CA**} & \multicolumn{4}{c}{**IRS**} & \multicolumn{4}{c}{**Ranking**} \\ \hline
**Image** & **at** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline  & \multicolumn{4}{c}{**STO**} & \multicolumn{4}{c}{**SCA**} & \multicolumn{4}{c}{**PSO**} & \multicolumn{4}{c}{**TO**} & \multicolumn{4}{c}{**DE**} & \multicolumn{4}{c}{**CA**} & \multic  
Table 5 presents the segmentation of the MRIs using EO for a qualitative inspection. From Figure 5, it is clear that two lumps in the prostate have been highlighted by the thresholding process. Prostatic MRIs present noisy conditions, which makes it difficult to visualize the thresholding with the naked eye, so in Figure 5 we present the thresholded image as well as the histogram with the values of the thresholds generated by the EO. It can be observed in the histogram that the thresholds present an adequate distribution, even though this particular image has impulsive noise and a simple shape.