## 3 Selection of Methods for Methodology
### Segmentation Algorithm Selection
Compared to conventional FCM, FRFCM offers simplicity and significantly improved processing speed, as it eliminates the need to calculate distances between pixels and clustering centers within local spatial neighborhoods. The proposed FRFCM method is executed on the grayscale histogram, resulting in the formulation of the objective function as follows:  
\[J_{FRFCM}=\sum_{l=1}^{q}\sum_{k=1}^{C}\gamma_{l}.u_{kl}^{m}\|\xi_{l}-v_{k}\|^{2} \tag{3}\]  
\(u_{kl}\) represents the fuzzy membership of gray value \(l\) concerning cluster \(k\); \(\sum_{l=1}^{q}\gamma_{l}=N\), and \(N\) is the total number of pixels in the grayscale image \(f=\{x_{1},x_{2},\ldots,x_{N}\}\), \(x_{1}\) is the gray value of the \(i\)-th pixel. \(v_{k}\) represents the prototype value of the \(k\)-th cluster. \(\xi\) is an image reconstructed by MR, and \(\xi_{l}\) is a gray level, (1\(\leq l\leq q\)); q denotes the number of the gray levels contained in \(\xi\). The \(\xi=R^{c}(f)\), where \(R^{c}\) is the morphological closing operation and \(f\) is the original image. The SFFCM method takes a novel approach by integrating the color histogram into its objective function, resulting in a faster segmentation process. The improved performance in color image segmentation is achieved by incorporating both adaptive local spatial information and global color features into the objective function:  
\[J_{SFFCM}=\sum_{l=1}^{q}\sum_{k=1}^{C}S_{l}.u_{kl}^{m}\|\left(\frac{1}{S_{l}} \sum_{p\in R_{l}}x_{p}\right)-v_{k}\|^{2} \tag{4}\]  
\(l\) (1\(\leq l\leq q\)) is the color level; \(q\) is the number of regions of the super-pixel image; \(C\) is the number of clusters; \(S_{l}\) is the number of pixels in the \(l\)-th region \(R_{l}\); \(x_{p}\) is the color pixel within the \(l\)-th region of the super-pixel image obtained by Multiscale Morphological Gradient Reconstruction-Watershed Transform MMGR-WT [42]. The GMM involves the superposition of multiple Gaussian distributions. The GMM distinguishes itself by its enhanced ability to handle outliers compared to K-means clustering. However, it does not consider spatial information [48], which may not effectively process content with noise or complex images. The GMM objective function can be defined as the log-likelihood of the data, which is to be maximized. For an image \(X=\{x_{1},x_{2},\ldots,x_{N}\}\) consisting of \(N\) data points (pixels) and a GMM with \(C\) Gaussian components, the objective function \(J_{GMM}\) can be defined as follows:  
\[J_{GMM}=\sum_{i=1}^{N}\log\left(\sum_{j=1}^{C}\pi_{j}\cdot N(x_{i}\left|u_{j},\ \sum_{j}\right.)\right) \tag{5}\]  
where \(N\) is the number of pixels, \(C\) is the number of Gaussian components in the mixture; \(\pi_{j}\) is the weight (or mixing coefficient) of the \(j\)-th Gaussian component, with the constraint \(\sum_{j=1}^{C}\pi_{j}=1\); \(x_{i}\) is the \(i\)-th pixel; \(\mu_{j}\) is the mean vector of the \(j\)-th Gaussian component; \(\sum_{j}\) is the covariance matrix of the \(j\)-th Gaussian component; \(N\) represents the probability density function of a multivariate Gaussian distribution with mean \(\mu\) and covariance matrix \(\sum\). The GMM's objective function seeks to find the parameters (\(\pi_{j}\), \(\mu_{j}\), \(\sum_{j}\)) that maximize the likelihood of the observed data. This is typically done using the Expectation-Maximization (EM) algorithm [49], which iteratively estimates the parameters by maximizing the objective function. In the E-step, the algorithm calculates the posterior probabilities of data points belonging to each Gaussian component. In the M-step, it updates the parameters to maximize the likelihood. The main requirement for the selection of segmentation methods was that the segmentation results created by the selected methods were visually distinguishable and provided different subjective evaluation results. Based on the segmentation tests using the AID, it was noticed that fuzzy subspace clustering (FSC), maximum entropy clustering (MEC),and FCM consistently produced visually very similar segmentation results.