## 6 Conclusions and Future Work  
In this paper, we introduced a novel approach aimed at improving Minimum Cross-entropy Thresholding (MCET) for image segmentation, particularly when dealing with images exhibiting skewed histogram distributions. Our primary objective is to enhance the estimation of the mean value in such cases. To achieve this, we leveraged the Maximum Likelihood Estimation of Gumbel distributions. We have utilized two variants of the Gumbel distribution, the maximum and minimum cases, and applied them to images with both right and left skewness, respectively. To validate our proposed method's effectiveness, we assessed the quality of the segmented images through both supervised and unsupervised evaluations, depending on the availability of ground truth data for which unsupervised evaluation is particularly valuable when segmenting images lacking corresponding ground truths. As can be seen from the results and their corresponding evaluations, our proposed method outperformed existing techniques that utilize MCET with alternative distribution models, highlighting our approach's efficacy in addressing the challenges of segmenting specific image types.  
In future research endeavors, we envision expanding our method's applicability by using heterogeneous skewness in image histograms and integrating it with diverse techniques to construct a versatile segmentation model. This extended model can be employed for various types of images and large datasets for segmentation and object detection purposes.  
This work was based on the efforts of the authors. W.A.H.J. and A.E.-Z.; supervision, A.E.-Z. Conceptualization, W.A.H.J. and A.E.-Z; methodology, W.A.H.J. and A.E.-Z.; software, W.A.H.J.; formal analysis, W.A.H.J. and A.E.-Z.; investigation W.A.H.J. and A.E.-Z; resources, W.A.H.J.; writing--original draft preparation and writing--review and editing, W.A.H.J.; visualization, W.A.H.J. and A.E.-Z. All authors have read and agreed to the published version of the manuscript.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Sentinel-2 Waterbodies**} \\ \hline
**Distributions** & **Avg. Evaluations** & **Improvement\%** \\ \hline Gaussian & 0.79126 & – \\ \hline Gamma & 0.81586 & +3.10\% \\ \hline Log-normal & 0.83740 & +5.83\% \\ \hline Gumbel—min & 0.82989 & +4.88\% \\ \hline Gumbel—max & 0.87041 & +10.0\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Improvement rates of the proposed method against different distributions when segmenting images with right-skewed histograms.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Dermoscopic Skin Lesion**} \\ \hline
**Distributions** & **Avg. Evaluations** & **Improvement\%** \\ \hline Gaussian & 0.78092 & – \\ \hline Gamma & 0.79089 & +1.27\% \\ \hline Log-normal & 0.79250 & +1.48\% \\ \hline Gumbel—min & 0.85891 & +9.98\% \\ \hline Gumbel—max & 0.80419 & +2.97\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Improvement rate of the proposed method against different distributions from segmenting images with left-skewed histograms.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** There was no ethical approval for this research since the authors used available datasets from public database, as stated in the manuscript.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available upon request.  
**Conflicts of Interest:** The authors declare no conflict of interest.