# Improving Prostate Image Segmentation Based on Equilibrium Optimizer and Cross-Entropy
## 1 Introduction
For industrial applications, the EO is applied to optimize the structural design of vehicles, specifically the vehicle's seat bracket [30]. The EO has been applied in the laser cutting process of polymethylmethacrylate sheets, where the EO enhances the accuracy of a random vector functional link network at estimating the laser-cutting properties of a given design [31]. In the field of artificial intelligence and machine learning, the EO has been used to train CNNs. In [32], the training process of a CNN involves the EO. The model is evaluated using real-time traffic data from IoT sensors in Minnesota, providing effective results. Even more, EO modifications have been applied to perform feature selection [33]. EO has been explored widely since its recent publication. For instance, Ref. [34] explores the effect of the incorporation of a chaotic map into the structure of EO with promising results. Following a similar idea, the authors also proposed an improved version of the EO by controlling the parameter \(t\)[35]. In [36], a mutation strategy was incorporated into the EO to modify the balance between exploration and exploitation. Lastly, EO was modified to address multi-objective problems in [37]. In [38] the authors proposed an EO-based thresholding approach for image segmentation using Kapur's entropy as an objective function over the Berkeley image segmentation dataset. Their results indicate that EO is able to outperform seven prominent metaheuristics algorithms, proving that EO is suitable for the segmentation task. Then, in [39], the authors presented an adaptive EO for image thresholding and tested their proposal with the BSDS 500 dataset. Given the above research, the relevance of EO in many areas is clear, making it a strong candidate for a large number of implementations. This evidence helped us to select the EO as the basis of the multilevel cross-entropy thresholding technique for the segmentation of prostate MRIs (MCE-EO). The proposed approach is designed to select an optimal set of threshold values using cross-entropy as the objective function. The proposed method is analyzed to evaluate its effectiveness over relevant images. The methodology contains two experiments. The first is designed to determine if the proposed MCE-EO is superior to related techniques over a set of generally used images in the field of thresholding, then we move towards a more in-depth analysis of the performance of MCE-EO on the specific domain of prostate image segmentation with magnetic resonance transaxial-cut prostate images. In both cases, the experiments consider six other metaheuristic algorithms: the sunflower optimization (SFO) algorithm [40], the sine cosine algorithm (SCA) [41], particle swarm optimization (PSO) [13], differential evolution (DE) [42], the genetic algorithm (GA) [43], and the Hirschberg--Sinclair algorithm (HS) [44]. The significance of the results is statistically analyzed, providing strong evidence of the suitability of MCE-EO for the segmentation of prostate MRIs. The article is structured as follows. Sections 2 and 3 present the theoretical foundations of the MCE-EO, including the formulation of cross-entropy in Section 2 and the description of the original EO in Section 3. Then, in Section 4 we propose the MCE-EO approach and all of its implications. Section 5 describes the experimental methodology and results. Finally, Section 6 concludes this article.