## 5 Results and Discussion  
For comparison purposes, we examined the proposed Gumbel-based MCET method and applied the same images as related works that used MCET with different distributions as a distribution-independent thresholding method. The performance measures were calculated by averaging the unsupervised and the supervised evaluations such that the best result among other results depended on the maximum evaluation. The images utilized depicted the necessary skewness for assessing the influence of employing the Maximum Likelihood Estimator (MLE) for the Gumbel distribution in both of its variants. These images were specifically chosen from their datasets as extreme cases, presenting significant difficulties when applying conventional image-segmentation techniques. As can be noticed in Figure 5, the proposed method has two main objective contributions in segmenting images with skewed histograms compared to related works. Using MCET with the maximum case of the Gumbel MLE showed obvious improvements over related works in segmenting Sentinel-2 Satellite images that demonstrate skewness to the right in their histograms; this applies to the concept of using the maximum Gumbel distribution case with right-skewness samples. Similarly, it is also noticeable that using MCET with the minimum case of the Gumbel MLE showed an improvement in segmenting dermoscopic skin lesion images that demonstrate skewness to the left in their histograms; this applies to the concept of using the minimum case of the Gumbel distribution with the left-skewness samples. While the Gumbel and log-normal distributions were employed to model the right distribution, the enhancement in segmentation achieved using the Gumbel maximum observed in the proposed method underscores the distinct characteristics of these distributions, which can vary depending on their specific parameters. Using the log-normal distribution in related works has the same concept of the maximum case in handling mean estimation for samples skewed to the right. Although the log-normal method shows an improvement against Gamma, the maximum case of the Gumbel distribution showed an additional improvement in this case; additionally, both cases showed slight improvements in the opposite side of the skewness regardless of their main objective improvement. As illustrated in Figure 6, the initial pair of images in column (a) depict Sentinel-2 satellite water bodies which exhibit right-skewed histograms. Conversely, the subsequent pair of images portrays dermoscopic skin lesions characterized by left-skewed histograms. As a subjective scene of segmented images, it is noticeable that the similarity of the segmented skin lesions and their reference images was very close when using MCET with the minimum case of Gumbel distribution. It is noticeable that these qualitative results cover  
Figure 5: The overall evaluation of the proposed MCET using the Gumbel distribution compared with the related methods’ different distributions. the object of interest in the skin lesions based on objects in the corresponding ground truth. As a quantitative result in Table 1 the corresponding average evaluation scores, 0.85880 and 0.83207, for images (a) 1 and 2, respectively, showed an obvious increase compared to other methods, representing the objective goal in this case and indicating that there is an enhancement in the mean value that positively impacts the objective function of MCET in the left-skewed histogram case. On the other hand, segmenting the Sentinel-2 images using the maximum case resulted in a similar subjective scene to the corresponding reference images. In Table 1, the evaluation scores, 0.84069 and 0.85816, for images (a) 3 and 4, respectively, reflect an objective improvement in the efficiency of the thresholding objective function based on the new estimated mean in the maximum case. Table 2 details improvement rates for the proposed method when segmenting right-skewed histograms of the Sentinel satellite water bodies using various distributions.