Identifications of Lung Cancer Using Kernel Weighted Fuzzy Local Information C-Means Algorithm

The proposed KWFLICM method for lung cancer diagnosis is discussed briefly in this section.

3.1 Lung cancer database

The Early Lung Cancer Action Program (ELCAP) and Lung Image Database Consortium and Image Database Resource Initiative (LIDC-IDRI) datasets are extensively used in lung cancer identification, particularly in computer-aided diagnosis (CAD) and detection systems. Here’s an in-depth look at each dataset’s contributions and specific features relevant to lung cancer identification:

3.1.1 ELCAP database

The ELCAP dataset is derived from a pioneering study focused on identifying lung cancer through Low-Dose CT scans (LDCT). Its purpose is to assist in early detection, particularly in populations at high risk, such as smokers. The key attributes for lung cancer identification are as follows:

• LDCT Scans: LDCT is a preferred method for lung cancer screening due to lower radiation exposure, making it safer for regular screening. However, LDCT presents challenges in maintaining high image quality, which tests the robustness of CAD algorithms to identify nodules despite lower contrast.
• High-Risk Population Focus: The scans are taken from individuals at high risk (e.g., heavy smokers) for lung cancer, providing data that simulates real-world screening conditions. This focuses on a high-risk demographic aid in developing algorithms more sensitive to early lung cancer signs.
• Detailed Nodule Annotations: Many CT images in ELCAP are annotated with nodule characteristics, including size, shape, location, and type (ground-glass opacities solid, or part-solid). These labels help algorithms learn to detect and classify lung nodules accurately.
• 3D Image Slices: ELCAP images are formatted in thin axial slices, allowing for volumetric (3D) analysis. This is particularly useful for distinguishing subtle nodules that may be hard to detect on single 2D slices but more evident when viewed across multiple slices.

The relevance’s of ELCAP for lung cancer identification are:

• Early Detection Focus: Since ELCAP aims to diagnose lung cancer, it is crucial for training and validating CAD algorithms focused on early-stage cancer identification.
• Volume Change Tracking: Some cases in ELCAP include follow-up scans, allowing for tracking of nodule growth over time. Growth rate is a significant indicator of malignancy, so this longitudinal data helps develop algorithms that can detect rapid growth patterns indicative of cancerous nodules.

3.1.2 LIDC database

• The LIDC dataset contains many CT scans and detailed nodule annotations. The key attributes of LIDC for lung cancer identification are as follows:
• Large Number of Cases: LIDC includes over 1,000 thoracic CT scans, providing a substantial dataset for training and testing lung cancer detection models.
• Multi-Rater Annotations: Each scan has been annotated by up to four radiologists independently, with each radiologist identifying nodule locations and providing detailed annotations. This multi-rater approach captures inter-observer variability, giving a realistic benchmark for CAD systems to match human performance.
• Detailed Nodule Characterization: Nodules larger than 3mm are annotated with several attributes, such as:
  • Nodule Diameter: Indicates the size of each nodule, with larger sizes potentially correlating with malignancy.
  • Sphericity and Spiculation: Shape attributes that help distinguish between malignant and benign. The former one often has irregular shapes and spiculated edges.
  • Subtlety: The subjective difficulty in detecting the nodule, allowing for testing of CAD algorithms’ sensitivity.
  • Malignancy Rating: Each radiologist rates the likelihood of malignancy on a scale, providing CAD systems with data on suspected malignancy levels to help them learn these distinctions.
  • Various Nodule Types: LIDC includes diverse nodule types. They can vary widely in appearance and malignancy risk. Nodule types included in LIDC are ground-glass, part-solid, and solid nodules. These samples are critical for training algorithms that can accurately segment and classify the different presentations of lung nodules.

The relevance’s of LIDC for lung cancer identification are:

• Malignancy Prediction: LIDC malignancy ratings allow algorithms to go beyond simple nodule detection and learn to assess the risk of cancer, making it useful for both detection (identifying nodules) and diagnosis (determining likelihood of malignancy).
• Inter-Rater Consistency Studies: The multiple annotations by different radiologists enable studies on how consistently different observers identify and classify nodules. This is essential for training CAD systems to handle the inherent uncertainty and variability in human annotation.
• Segmentation and Classification Benchmarks: Since LIDC provides rich, detailed annotations, it is commonly used as a benchmark for testing the accuracy of segmentation algorithms in delineating nodule boundaries and classifying nodules as malignant or benign.

3.2 Image segmentation and image engineering

The process of creating a section of an image or object is known as segmentation. Pattern recognition and image analysis are the first steps in image segmentation. Segmenting videos with dynamic backgrounds is an almost essential area of research in computer vision. Most functions related to image processing and analysis include evaluating or assessing images. The technique of splitting an image into separate, homogeneous, or somewhat similar sections is known as image segmentation. Consequently, characterization, delineation, and visualization of the region of interest in each image are critical components of image segmentation.

Depending on whether a pixel's intensity value is larger or lower than a predetermined threshold, the image is separated into white and black pixels. Three layers of image engineering are shown in Figure 1. The upper layer is used for image understanding, the middle layer is used for image analysis, and the bottom layer is used for image processing. The result of image segmentation is significantly influenced by the precision of feature measurement. As image segmentation is a computer-aided process, the automation of segmentation is essential for medical imaging applications. The image segmentation and image engineering application areas are depicted in Figure 1.

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Figure 1. Image engineering and image segmentation

3.3 FCM algorithm

The fuzzy clustering and hard clustering are two clustering strategies. Usually, just one cluster is assigned to each data point using hard clustering techniques. As a result, every pixel in an image belongs to one cluster, yielding extremely accurate results in image clustering. The effectiveness of hard clustering algorithms is diminished by several factors, including the intensity of homogeneities, the lack of contrast, the insufficient spatial resolution, overlap, and noise. The fuzzy clustering technique has been extensively researched and used to the process of image clustering and segmentation. It is a soft segmentation approach because it can withstand ambiguity and maintain more information than hard segmentation techniques. The FCM approach is used more often than other fuzzy clustering algorithms. This is among the reasons why it is so popular. For determining the optimal division, iterative FCM clustering is used to minimize the weighted within-group sum-of-squared error objective function.

where, $Y=\left\{Y_1, Y_2, \ldots Y_M\right\} \subseteq I R^m$ data that is stored in m-dimensional vector space, M consists of the total amount of data pieces, d refers to the total number of clusters that $2 \leq d<M$, $u_{a b}$ is the extent to which one is a member of $Y_{a}$ in the $a^{t h}$ cluster, p the weighting exponent that is applied to each fuzzy membership status, $V_{b}$ is an example of the center of cluster prototype iteration (b), $d^2\left(Y_a, V_b\right)$ a measurement of the distance between two objects $Y_{a}$ and cluster center $V_{b}$. An answer to the problem of the object function $K_{p}$ attainable using an iterative procedure, which is carried out in the following manner:

(1) Values for d, p and $\epsilon$ are set.

(2) Initialize the Fuzzy partition matrix $U^{(0)}$.

(3) Set the Loop counter b as b=0.

(4) Compute the Cluster centers $V_b{ }^{(b)}$ with $U^{(b)}$ using Eq. (2):

$V_j^{(b)}=\frac{\sum_{i=1}^N\left(U_{j i}(b)\right) m x i}{\sum_{i=1}^N\left(U_{j i}(b)\right) m}$                          (2)

(5) Compute the membership matrix suing Eq. (3):

$V_j^{(a+1)}=\frac{1}{\sum_{k=1}^c\left(\frac{d_{j i}}{d_{k i}}\right)^{\frac{2}{n+1}}}$                         (3)

(6) If max $\left\{U^{(b)}-U^{(b+1)}\right\}<\epsilon$ then stop, otherwise, set $a=a+1$ and go to step 4.

Figure 2 presents a comparison of the clustering results obtained by the FCM after applying it to a synthetic test image.

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Figure 2. (a) Noisy image; (b) FCM result

3.4 FLICM algorithm

One novel and trustworthy FCM framework developed in this paper is the Fuzzy Local Information C-Means (FLICM) method. Although the approaches discussed in the preceding section produced satisfactory results for image clustering, they do have certain shortcomings. They aren't strong enough to withstand noise and anomalies, particularly when the noise is unknown in advance, even if using local spatial information makes them more noise resistant. A key parameter, denoted as $\alpha$ (or $\lambda)$, is incorporated into their objective functions to strike a balance between the efficacy of image detail preservation and resistance to noise. Choosing it usually requires some trial and error or expertise. It is expected that the new component will possess several unique qualities:

• For preserving robustness and noise insensitivity via the use of fuzzy integration of local spatial and local intensity information.
• To control the influence of neighboring pixels in direct proportion to the distance between them and the anchor pixel.
• In order to preserve all of the nuances of the initial image without making any preparations before the process. Consequently, the new fuzzy factor is defined as the following:

$H_{n m}=\sum_{p \in M_m}^m \frac{1}{e_{m p+1}}\left(1-v_{n p}\right)^0\left\|Y_p-U_n\right\|^2$                        (4)

where, the $m^{t h}$ local window center is denoted by the pixel, reference cluster is referred as n, $p^{t h}$ it is recommended that the pixel be included in the accumulation of neighbors falling into a window inside the system, the $m^{t h}$ pixel $\left(M_m\right)$, $d_{i j}$ represents as the spatial Euclidean distance between each pair of pixels i and j, $u_{k i}$ means the extent to which one is a member of the $p^{t h}$ pixel in the $n^{t h}$ cluster, O is the weighted exponent that is applied to each fuzzy membership, and $U_{n}$ is the model of the center of cluster that is developed. n is a significant role played by $H_{n m}$ the application of the method will also be shown in the part that comes after this one. By using the definition of $H_{n m}$ the FLICM clustering technique is described as a robust framework for image clustering that utilizes fuzzy convolutional neural networks. It includes information about the local spatial and intensity level into its objective function, which is expressed in terms of $J_{m}$ in below equation.

$J_m=\sum_{m=1}^M \sum_{n=1}^d u_{n n}^{\circ}\left\|Y_p-U_n\right\|^2+H_{n n}$                          (5)

The two necessary conditions for $J_{m}$ to be at its local minimal extreme, with respect to $u_{n m}$ and $u_{m}$ is obtained as follows:

$u_{k i}=\frac{1}{\sum_{j=1}^c\left(\frac{\left\|X_i-V_k\right\|^2+G_{k i}}{\left\|X_i-V_j\right\|^2+G_{j i}}\right)\left(\frac{1}{m-1}\right)}$                        (6)

$V_k=\frac{\sum_{i=1}^N u_{k i} m_{x i}}{\sum_{i=1}^N u_{k i} m}$                          (7)

Thus, the FLICM algorithm is given as follows:

(1) Initialize the fuzzification parameter (m), cluster prototypes (c), and the stop condition $\epsilon$.

(2) Initialize the Fuzzy partition matrix.

(3) Initialize the Loop counter b as b=0.

(4) Compute c using Eq. (6).

(5) Compute the Membership values using Eq. (7).

(6) $\max _{i j}\left\{U^{(b)}-U^{(b+1)}\right\}<\epsilon$ then stop, otherwise, set b=b+1 and go to step 4.

Figure 3 shows the comparison of clustering results on synthetic test image by the FCM and FLICM techniques.

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Figure 3. (a) Original image; (b) FCM Result; (c) FLICM Result

3.5 Proposed KWFLICM algorithm

The proposed KWFLICM algorithm enhances image segmentation by accurately detecting irregular shapes and boundaries of lung cancer tissues. This section shows how KWFLICM improves upon traditional clustering techniques like FCM and FLICM, especially in lung cancer detection. The challenges in lung cancer identification are as follows:

• Irregular Shapes: Lung cancer lesions often have irregular shapes and textures, making them difficult to capture with simple clustering.
• Noise and Artifacts: Medical imaging can introduce noise, especially in complex areas like lung tissue.
• Fuzzy Boundaries: The boundary between cancerous and non-cancerous tissue can be indistinct, so precise boundary detection is critical.

KWFLICM addresses these challenges by integrating kernel methods with spatial weighting, making it especially effective for lung cancer image segmentation. FCM lustering algorithm calculates a membership value for each pixel based solely on pixel intensity, which limits its ability to handle noise and fuzzy boundaries. FLICM incorporates spatial information from neighboring pixels to improve noise resilience, but it may still struggle with complex textures and irregular boundaries in lung images. KWFLICM improves upon these by using kernel functions and weighted local spatial information, enabling better differentiation of cancerous regions. The key enhancement in KWFLICM is the kernel function, which transforms data to a higher-dimensional space. This allows KWFLICM to:

• Capture Nonlinear Patterns: The higher-dimensional representation makes it easier to capture complex relationships and patterns within lung tissues, often irregular in shape.
• Differentiate Lung Nodules: The transformation allows KWFLICM to differentiate small, irregularly shaped nodules and other lung cancer features from normal tissue more effectively.

KWFLICM uses a weighted local spatial information mechanism that considers both the intensity and spatial proximity of neighboring pixels. This helps in:

• Enhancing Noise Robustness: Weighted spatial information helps distinguish noise from actual cancerous patterns, crucial in lung CT scans where noise or artifacts can obscure critical details.
• Improving Boundary Accuracy: KWFLICM applies spatial weighting in the transformed kernel space, allowing it to more accurately detect fuzzy boundaries typical of lung cancer lesions.

The KWFLICM algorithm optimizes an objective function that minimizes:

• Kernelized Distance: The kernel function transforms pixel values, making it easier to distinguish clusters in a higher-dimensional space.
• Weighted Local Information: Local spatial weighting accounts for neighboring pixel similarity, making clustering more accurate and reducing noise interference.

The KWFLICM algorithm follows these steps in lung cancer identification:

• Initialization: Initial cluster centers and memberships are set.
• Kernel Transformation: To map the data to a higher-dimensional space, each pixel’s intensity is transformed via the kernel function.
• Membership Update: Membership values are computed based on kernelized distance and weighted local information.
• Cluster Center Update: Centroids are updated by averaging the membership-weighted pixel values.
• Iteration: The process repeats until the objective function converges, stabilizing cluster assignments.

In lung cancer identification, KWFLICM's combination of kernel functions and weighted spatial information improves the algorithm’s ability to handle noise, detect complex boundaries, and capture small, irregular cancerous regions. This enhances the accuracy of segmentation, making KWFLICM a powerful tool for identifying and analyzing lung cancer in medical imaging. Kernel parameters are typically chosen in the KWFLICM algorithm for lung cancer identification: The choice of kernel function is as follows:

• Common Kernel Types: The Gaussian (RBF) kernel and polynomial kernel are frequently used in KWFLICM due to their adaptability for medical imaging.
• Gaussian (RBF) Kernel: Suitable for capturing non-linear relationships within lung tissues, this kernel is popular in medical imaging because it can distinguish between subtle intensity variations.
• Polynomial Kernel: Effective in separating clusters when the structure is less complex; higher-degree polynomials can capture more intricate patterns.
• Selection Process: The choice is often based on trial experiments. For instance, Gaussian kernels are generally preferred when lung lesions have complex shapes and textures, while polynomial kernels are considered if the relationship between cancerous and normal tissue regions appears relatively linear.

The RBF kernel parameter (σ) controls the smoothness of the Gaussian curve, influencing the radius of influence for each pixel’s similarity with neighboring pixels. The optimal values for σ are often found through cross-validation by comparing different values on a set of sample images, selecting the one that achieves the best segmentation accuracy. For lung cancer images, σ is usually chosen based on pixel intensity ranges and the image resolution, as lower σ values capture finer details, while higher σ values generalize more. The degree (d) in polynomial kernel controls the complexity of the polynomial surface. Higher degrees can capture more complex relationships but can also lead to overfitting. The coefficient (c) shifts the polynomial surface, controlling the sensitivity of the kernel function to pixel intensity differences. Typically, the values of d range from 2 to 4 ensure the polynomial function is neither too simple nor too complex for the lung images. The coefficients are manually adjusted based on visual inspection of segmentation results; high values may enhance contrast but also risk introducing artifacts.

The fuzzy clustering approach is the most essential clustering method for image segmentation since it can preserve more information about the image than the hard clustering method. The FCM approach is a popular fuzzy clustering method for image segmentation. Many considerations must be satisfied for a fuzzy clustering method to function as intended. Concerns about the initialization of the clustering method, cluster size and form, cluster distribution, cluster number in the data, and cluster number are at the heart of these problems:

$J_m=\sum_{k=1}^M \sum_{l=1}^N u_{k i} m\left\|Y_k-A_l\right\|^2, 1 \leq m<\infty$                         (8)

where, $u_{i j}$ is the degree of membership of $Y_{k}$ in the cluster k, m is any real number larger than 1, the d-dimension center of the cluster is $A_{l}$, along with that $\left\|Y_k-A_l\right\|$ represents any norm that expresses the degree of similarity between a set of measured data and the center of the set and $i^{t h}$ of d-dimensional measured data is $Y_{k}$. The process of fuzzy partitioning may be carried out by performing an iterative optimization of the objective function that is provided before, with the status of membership being updated during the process. $u_{i j}$ and the cluster centers $A_{l}$ by:

$v_{k l}=\frac{1}{\sum_{p=1}^a\left(\frac{\left\|Y_k-A_l\right\|}{\left\|Y_k-A_p\right\|}\right)^{\frac{2}{n-1}}}$                      (9)

$A_j=\frac{\sum_{i=1}^N u_{i j}^m x_i}{\sum_{i=1}^N u_{i j}^m}$                          (10)

After $\max _{i j}\left\{\left|u_{i j}^{(k+1)}-u_{i j}^k\right|\right\}<\epsilon$, the iteration will stop, where, $\epsilon=[0,1]$ a termination criterion and the iteration steps are denoted by k. This process converges with a saddle point of $J_{m}$ or local minimum. When it comes to noise-free images, the FCM methods outperform more traditional algorithms. However, images contaminated with noise, outliers, and other imaging aberrations cannot be segmented using this method. FCM generates findings that are not robust for several reasons, the most significant of them are the use of a Euclidean distance and the absence of spatial contextual information in the image that is not robust. Many other FCM algorithms that have been improved have been offered in attempt to find a solution to the first difficulty. Each of these algorithms incorporates local geographical data into the FCM goal function that is first developed. Within the framework of the FCM_S algorithm, the spatial neighborhood term is included to bring about a modification to the mission function of FCM.

One of the most significant downsides of FCM_S is that the computation of the spatial neighborhood term requires a significant amount of time during each iteration step. This presents a significant challenge for the algorithm. It is decided to construct two distinct versions of FCM_S, namely FCM_S1 and FCM_S2, to lessen the amount of computing effort that is needed by FCM_S. To improve the overall performance of the model, the neighborhood term of FCM_S is changed to a median-filtered image in FCM_S2, while an additional mean-filtered image is included into FCM_S1 to further improve its performance.

The capacity to calculate the mean- and median-filtered images in advance results in a reduction in the operating expenses associated with processing. Enhanced FCM (EnFCM) is created for accelerating the process of image segmentation. For producing a linearly weighted sum image, it makes use of the original image as well as the average grey level of each pixel in the surrounding neighborhood. The combined image's grey level histogram is then used for categorization. The computational complexity of EnFCM is less. However, these methods do not allow for the direct application of the source image. To manage the balance between robustness against noise and detail preservation efficacy, they require certain parameters, denoted by an or λ. To address these issues and improve image segmentation performance, a new parameter-free robust FLICM is introduced. By defining a new fuzzy factor, FLICM supplants parameter a, which has been used by all prior methods and variations.

One of the objectives is to develop a trade-off WFF that is capable of being used for the purpose of adaptively controlling the local spatial connection. The distance that separates neighboring pixels and the difference in intensity levels between them are the two factors that define this section. To enhance the FLICM method, a trade-off WFF and kernel approach are employed. Both the intensity and the spatial distance between adjacent pixels are considered by the tradeoff WFF. Moreover, there are no parameters for the kernel distance measure, or the tradeoff WFF. Following segmentation using the FCM, FLICM, and KWFLICM algorithms, the tumor region from the segmented image is shown separately. Figure 4 displays the algorithm's flow chart.

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Figure 4. Flow chart representation of KWFLICM algorithm

The following is a concise summary of the algorithm that has been proposed:

• Define c as the number of cluster prototypes, $N_{i}$ is window size, m fuzzification parameter and the stopping condition.
• The fuzzy cluster prototypes should then be initialized in a random manner.
• Loop counter b = 0 is set after this.
• To determine the changed distance measurement, calculate $D_{i k}^2$. The weighted fuzzy component of the trade-off $W_{i j}$.
• Adjust the cluster prototypes and the partition matrix as necessary.

If $max \left|V_{ {new }}-V_{ {old }}\right|<\epsilon$ then stop, otherwise, set b = b+1 and go to step 4. $V=V_1, V_2, \ldots V_c$ are vectors that represent the prototypes of the clusters. After the algorithm has reached a point of convergence, the defuzzification procedure is carried out to provide a segmented image that is clear and distinct.

3.6 Membership function

Fuzzy sets' Membership Function (MF) is a simplified version of classical sets' indicator function. As an expansion of value, the MF in fuzzy logic stands for the degree of truth. Although they are conceptually different, degrees of truth and probabilities are sometimes used interchangeably. Fuzzy truth, on the other hand, denotes membership in nebulously defined sets rather than the probability of a certain occurrence or situation. A membership function for a fuzzy set A on the universe of discourse X is denoted as: X $\rightarrow$ [0,1], where each element of X is mapped to a value in between to 0 and 1.

3.6.1 Cluster prototype

Fuzzy modeling, data analysis, image processing, pattern recognition, and FCM algorithms are just a few of the applications that have made extensive use of fuzzy clustering approaches based on objective functions. Using fuzzy clustering techniques, it is feasible to ascertain an underlying structure that exists within the data. The data set is divided into groups that overlap with one another, which is one instance of these methodologies. To obtain better results, a technique that employs fuzzy clustering must consider a variety of distinct elements. The configuration of the clustering method, the number of clusters present in the data, the form and size of the clusters, and the distribution of the data pattern should all be taken into consideration. Computing the morphologies of the clusters is accomplished via the use of methods that combine point prototypes with the distance measure. Data points from linked clusters with membership 1.0 should be included in the cluster prototypes, which should also extend a certain distance from the cluster centers, according to this suggestion. This is not a characteristic of current clustering methods. The sensitivity of fuzzy clustering algorithms to initialization is well-known. It is common practice to randomly start the algorithms numerous times in the hope that one of them would yield a good clustering result. A very diverse distribution of data patterns makes the system more sensitive to startup.

3.6.2 Trade-Off weighted fuzzy factor

The noise resistance asset of the proposed KWFLICM mainly relies on the fuzzy factor $H_{m n}$ as shown in below equation.

$H_{m n}=\sum_{i=1}^N \sum_{k=1}^c u_{m n}^m \sum_{i \neq j}^N V_{a b}\left(1-V_{m n}\right)^m\left(1-J\left(Y_a, U_m\right)\right)$                      (11)

To what extent the adaptive trade-off WFF considers local intensity-level and geographical constraints are context dependent. The spatial constraint is defined as the amount to which neighboring pixels with respect to spatial distance from the center pixel (in terms of both $p_{i}$ and $q_{i}$) dampen the effect of pixel $x_{i}$.

$W_{s c}=\frac{1}{d_{i j}+1}$                          (12)

where, the $i^{t h}$ local window center is denoted by the pixel $N_{i}$ and the $j_{t h}$ pixel is a representation of the collection of neighbors that have fallen into the window surrounding it the $i^{t h}$ pixel, $d_{i j}$ refers to the Euclidean distance between the two points in space $j^{t h}$ the pixel at the center, as well as the neighboring particles. The impact of the pixels included inside the local window may change in a flexible way according to their distance from the center pixel because of the specification of the spatial component. This is because the local window is a locally contained area. Because of this, it is possible to make use of extra information that is in the immediate vicinity. Table 1 shows a 3×3 layout that depicts the window with noise as well as the dampening extent of the neighbors.

Table 1. Noise and the amount to which neighbors dampen it are shown in a 3×3 window (a) No noise in the central pixel; (b) Noisy pixels have ruined the center pixel; (c) The surrounding area's level of moisture

An instance of this may be seen in Table 1(a), which shows a 3×3 window taken from the noise image, and Table 1(c) shows how its neighbors' damping extent varies with spatial distance. By including the fuzzy factor, Table 2 illustrates the changes in membership values corresponding to Table 1(a). Table 2 shows that when the iteration counts get up, the membership values of noisy and no-noisy pixels converge on a common value, and the noisy pixels are ignored. The method converges after five rounds. In this scenario, the intensity level values of the noisy pixels are distinct from those of the other pixels included inside the window, while the fuzzy component contributes to the overall pattern ($G_{k i}$) balances their own membership values. Every single pixel that is included inside the window is a member of the same cluster. Accordingly, the factor incorporates both the intensity level and the spatial limitations, which together form the combination $G_{k i}$, the impact of the noisy pixels should be suppressed.

Table 2. Membership values (a) Initial membership values; (b) After a single phase; (c) Subsequent to two iterations; (d) three times through the process; (e) Subsequent to four iterations; (f) Similar membership values

The element $G_{k i}$ is also decided automatically rather than being set synthetically. It does not know any information about the noise. Hence, the algorithm becomes more resistant to the outliers being used. Following that, the coefficient of variation at the local level $C_{j}$ for each pixel j as $C_j=\frac{{var}(x)}{x^2}$ is obtained. $C_{j}$ displays how consistent the intensityscale of the window that is around it is. Because of this, the values are low in regions, while the values are high at edges or in locations where there is noise corruption. According to the data point distance variance, the degree of aggregation that exists around the clusters may be determined. Although there is just a little amount of variance, the clusters are somewhat near to one another and are spread out around their cores. To restate, it is envisaged that membership will be shown separately if the dataset contains recognizable clusters. Members will be exhibited individually.