## 3 Selection of Methods for Methodology
### Segmentation Algorithm Selection  
The process of selecting an image segmentation algorithm involves considering various factors and criteria like dataset characteristics and application area. To demonstrate the practical realization of methodology, we selected five soft clustering alternatives: fuzzy C-means (FCM) [41], superpixel-based fast fuzzy C-means (SFFCM) [42], significantly fast and robust fuzzy C-means (FRCM) [40], robust self-sparse fuzzy clustering (RSSFCA) [43], and the Gaussian mixture model (GMM). Except for the GMM, all segmentation algorithms are modified versions of the FCM segmentation method. Fuzzy image segmentation algorithms can handle uncertainties and ambiguities in the image data. The original FCM clustering algorithm has been widely applied for image segmentation problems. FCM allows each pixel of an image to belong to multiple clusters simultaneously, which is useful for images with complex textures and overlapping objects. However, FCM has limitations such as sensitivity to noise, initialization dependency, and slow convergence, which led to the development of many modified versions of FCM such as T-spherical fuzzy C-means [44], improved fuzzy subspace clustering [45], and an adaptive entropy weighted picture fuzzy clustering algorithm with spatial information (APPCM_S) [46]. Image segmentation algorithms, including clustering-based methods like FCM, define the objective function that serves as a mathematical representation of the goal the algorithm aims to achieve. The objective function quantifies how well the segmentation algorithm is performing and helps guide the algorithm to find the optimal segmentation solution. The objective function is often designed to be minimized. The classical FCM algorithm defines its objective function as follows:  
\[J_{FCM}=\sum_{i=1}^{C}\sum_{j=1}^{N}u_{ij}^{m}.d_{ij}^{2} \tag{1}\]  
\(J_{FCM}\) is FCM's objective function that aims to minimize the distance between pixels and cluster centers while considering the extent to which each pixel belongs to each cluster. The parameter \(m>1\), referred to as the fuzzifier, controls the level of fuzziness in the clustering outcome, with higher \(m\) values resulting in more fuzzy clusters. \(C\) is the number of clusters, and \(N\) is the number of pixels. \(d_{ij}\) is the distance from the \(j\)-th pixel to the \(i\)-th cluster. \(u_{ij}\) is the degree of membership of the \(j\)-th pixel in the \(i\)-th cluster \(\left(0\leq u_{ij}\leq 1\right)\). Researchers often modify objective functions depending on the image content to enhance segmentation results. A common solution to suppress the influence of noise is the inclusion of (fuzzy) local spatial information inside the objective function [47]. Traditional fuzzy clustering algorithms encounter several problems. They are sensitive to outliers due to the non-sparse nature of fuzzy memberships. Secondly, these algorithms often lead to excessive image over-segmentation, primarily due to the loss of local spatial information within the image. These issues are solved by the RSSFCA method. The authors define a final objective function in the following form:  
\[J_{RSSFCA}=\sum_{i=1}^{C}\sum_{j=1}^{N}u_{ij}.\Phi^{\prime}\left(x_{j}|v_{i},\ \sum_{i}\right)+\gamma.\sum_{i=1}^{C}\sum_{j=1}^{N}u_{ij}^{2}. \tag{2}\]  
\(\Phi^{\prime}\left(x_{j}|v_{i},\ \sum_{i}\right)\) represents the distance metric between \(x_{j}\) and \(v_{i}\) while also ensuring the requirement of positive values on the distance measure metric. The \(x_{j}\) denotes the \(j\)-th pixel in the image and \(v_{i}\) is the \(i\)-th clustering center. Parameter \(\gamma\) is a balancing parameter that controls the sparsity level within memberships. Adjusting the \(\gamma\) value results in variations in the objective function, leading to different levels of resilience to outliers or noise. FRFCM represents a significant advancement in FCM algorithms, utilizing morphological reconstruction (MR) and membership filtering to enhance speed and robustness.