Edge Detection Using Fuzzy C-Means: A Comparative Study

Edge detection is a critical step in image processing, serving as a precursor to tasks like feature extraction, object recognition, and segmentation. It simplifies images by reducing data while preserving essential structural details. Traditional techniques like Sobel, Prewitt, and Roberts often face challenges such as noise sensitivity and thick edge lines. Fuzzy clustering methods, particularly Fuzzy C-Means (FCM), offer an alternative by leveraging probabilistic boundaries to handle overlapping regions effectively. This study aims to evaluate the performance of FCM in edge detection and compare it with conventional methods [1] [4] [6]. The Fuzzy C-Means (FCM) clustering algorithm offers a promising alternative for addressing these challenges. Unlike traditional methods, FCM leverages fuzzy logic to handle overlapping regions and ambiguities in image data, making it particularly effective for tasks that require precise segmentation. This study explores the integration of FCM into the edge detection process, highlighting its potential advantages over conventional techniques. By focusing on the detection of brain tumor edges in MRI images, the research underscores the practical significance of FCM in medical imaging, where accurate and reliable analysis is crucial for diagnosis and treatment planning. This paper aims to provide a comparative analysis of FCM-based edge detection with traditional methods, using MATLAB as the implementation platform. The investigation covers key performance metrics, including accuracy, noise resistance, and computational efficiency, to offer a comprehensive evaluation of each approach. The findings contribute to the growing body of knowledge in image processing and provide valuable insights for researchers and practitioners working in fields such as medical imaging, computer vision, and machine learning.

1. Methodology

Figure 1 Edge Detection

1. 1. Traditional Edge Detection Techniques

Canny edge detection is a widely used method due to its precision and effectiveness in identifying edges within an image. The process consists of the following key steps [2]:

1. Noise Reduction: A Gaussian filter is applied to smooth the image and reduce noise, ensuring that minor variations do not get mistaken for edges.

• Gradient Calculation: Gradients in the image are calculated to detect regions with significant intensity changes, marking them as potential edges.
• Edge Thinning (Non-Maximum Suppression): The algorithm refines detected edges by suppressing non-maximum gradient values, producing a clearer and more accurate edge map.
• Edge Linking (Hysteresis Thresholding): Strong edges are retained, and weaker edges are connected based on a dual-threshold technique to form continuous boundaries.
• Advantages: This method is effective in detecting well-defined edges and produces fewer false positives in clean images. It is extensively used in computer vision tasks such as object detection, image segmentation, and medical image processing.

2. Sobel Operator: The Sobel operator is a gradient-based edge detection technique that computes the gradients of image intensity in both horizontal and vertical directions using specific convolution masks [4].

• How it works: The operator applies a pair of 3x3 convolution kernels to detect edges in the x and y directions. The gradients are then combined to obtain the overall edge strength.
• Key Benefits: It highlights regions in the image where there is a significant contrast in intensity, making it effective in detecting both horizontal and vertical edges [13].
• Efficiency: Sobel is computationally efficient compared to more complex methods like Canny, as it requires fewer calculations and works in real-time applications.
• Applications: Sobel is commonly used in image processing tasks, including object recognition, motion detection, and visual analysis for autonomous vehicles, where fast processing is required.

3. Prewitt Operator: The Prewitt operator, like Sobel, detects edges by computing image gradients using convolution masks. However, it differs in the design of its kernels [13].

• Gradient Calculation: Prewitt uses a pair of convolution masks that are symmetric in their gradient calculations, applying similar operations for both horizontal and vertical edge detection [14].
• Comparison to Sobel: While similar in function, Prewitt is generally simpler and often produces less precise results when dealing with noisy images due to its symmetry and lack of a weighting mechanism for diagonal edge directions.
• Simplicity: The Prewitt operator is less computationally intensive than Sobel, making it suitable for applications where simplicity and low computational cost are more important than edge precision.
• Applications: Prewitt is used in applications where simplicity and speed are priorities, such as basic image preprocessing or systems with limited resources.
  1. 1. Edge Detection

Edge maps can be generated using various algorithms, including Roberts, Prewitt, Sobel, and more advanced techniques like LoG and Canny. The effectiveness of these methods largely depends on the characteristics of the original image. Enhanced images often exhibit multiple levels of intensity gradation, which can result in the detection of false edge fragments during edge detection. To address this, a preliminary segmentation step using the Fuzzy C-Means (FCM) clustering method was employed. The Fuzzy C-Means clustering technique identifies a set of fuzzy clusters and corresponding cluster centers that best represent the data structure. This method divides a dataset of size n into a specified number of fuzzy clusters. A key aspect of FCM is the fuzzy membership matrix W={w_ik}, where each element w_ik ? indicates the degree to which the k-th data point belongs to the i-th cluster. For a given number of clusters c, FCM partitions the dataset X= {x₁, x₂,…,x_n} into c fuzzy clusters with cluster centers V={v₁,v₂,…,v_c}, while minimizing the objective function,

Here, m represents the fuzziness index, w_i?k denotes the degree of membership of the data point x_k? to the i-th cluster, v_i is the center of the i-th cluster, and ?x_k−v_i?² indicates the squared distance between the data point x_k? and the cluster center v_i?.

Figure 2 (a) Original Image, (b) Enhanced Image, (c) Result After FCM Clustering, (d) Final Edge Map

In each iteration of the FCM clustering algorithm, the membership matrix W is updated using Equation (w_ik), and the cluster centers are recalculated using Equation (V_i). The square error is then computed using Equation F (W, V). The algorithm terminates when the error falls below a predefined tolerance value or when the improvement between consecutive iterations is less than a specified threshold. The parameter mmm plays a key role in determining the influence of membership grades in the performance index. As mmm increases, the partition becomes more diffuse, and in the limit as m→∞, all objects will belong to all clusters with equal membership degrees. Furthermore, mmm impacts the calculation of cluster center coordinates by amplifying the influence of objects with higher membership values and reducing the impact of those with lower membership values. After the image is segmented into a set of homogeneous regions using the FCM clustering algorithm, the Canny edge detector is applied to identify edges. This method utilizes the gradient value of each pixel to detect fine edges while maintaining the homogeneity of the image regions. The process involves several steps. First, a Gaussian filter is applied to smooth the original image and reduce noise. Next, the gradient magnitude and direction are calculated for each pixel. Then, boundary pixels (edge pixels) are identified by comparing the gradient magnitude of each pixel to those of its two neighbors in the gradient direction. A pixel is classified as a boundary pixel if its gradient magnitude exceeds that of its neighbors.

1. 1. 2. Applications of Edge Detection

Edge detection is widely used in various applications across computer vision and image processing. In object recognition, it helps define the boundaries of objects, making them easier to identify and classify. For image segmentation, edge detection serves as a critical step in dividing an image into distinct regions of interest, which is crucial for applications like medical image analysis and object detection. In image enhancement, it improves the visibility of objects by emphasizing their boundaries. Additionally, edge detection plays a significant role in pattern recognition by identifying structures and patterns in images, such as in fingerprint analysis or facial recognition systems. These applications highlight the importance of edge detection in simplifying complex image data and enabling more accurate analysis and interpretation [3] [6] [17].

1. 2. Fuzzy C-Means Clustering

Fuzzy C-Means is an advanced clustering algorithm used to partition image pixels based on their intensity values and spatial relationships [8] [9] [14]. Unlike hard clustering methods where a pixel belongs to a single cluster, FCM allows partial membership in multiple clusters, which helps in handling data ambiguity (Figure 3). The FCM algorithm involves:

1. Initialization: Randomly initialize cluster centers.
2. Membership Assignment: Each pixel is assigned a membership value representing its likelihood of belonging to a cluster.
3. Cluster Center Update: Recalculate cluster centers by weighting pixel values based on their memberships.
4. Iteration: Repeat the above steps until the changes in membership values and cluster centers are below a predefined threshold [6][18].

Figure 3 Fuzzy C-Means Clustering Algorithm

Where

• μ_ij?: Membership degree of pixel (or data point) x_j in cluster c_i.
• m: Fuzziness parameter (controls the fuzziness of clustering, typically m > 1).
• ?x_j−c_i?²: Squared Euclidean distance between the data point x_j? and the cluster center c_i?.
• c: Number of clusters.
• N: Total number of data points.

This iterative process enables FCM to produce fine edges by identifying and clustering pixels with similar intensity and spatial characteristics.

2. Operations on Fuzzy
   1. Basic Fuzzy Set Operations

These operations extend classical set theory by incorporating the concept of membership degrees, which range between 0 and 1.

• Union: The union of two fuzzy sets, A and B, combines the maximum membership values of each element in the sets:

• Intersection: The intersection of A and B takes the minimum membership values of each element:

• Complement: The complement of a fuzzy set A determines how far each membership degree is from 1:

1. 2. Fuzzy Relations

Fuzzy relations describe associations between elements of fuzzy sets.

• Composition of Relations: Given two relations, R (between sets X and Y) and S (between sets Y and Z), their composition determines the combined relationship between X and Z. It is computed using the max-min rule:

1. 3. Aggregation Methods

Aggregation combines fuzzy sets or their membership degrees into a single value.

• Algebraic Sum: The algebraic sum merges memberships while preventing the total from exceeding 1:

• Algebraic Product: This operation multiplies the membership values of two fuzzy sets:

1. 4. Arithmetic Operations on Fuzzy Numbers

Fuzzy arithmetic enables operations like addition, subtraction, multiplication, and division on fuzzy quantities.

• Addition: The sum of two fuzzy sets is calculated by maximizing the minimum memberships of their combined values:

• Multiplication: The product of fuzzy sets uses a similar approach:

1. 5. Defuzzification

Defuzzification is the process of converting a fuzzy set into a crisp value, commonly used in decision-making.

• Centroid Method: This calculates the center of gravity of a fuzzy set:

• Mean of Maximum: This method averages all values where the membership is at its peak.
• Maximum Membership: It selects the element with the highest membership degree as the crisp output.
  6. Fuzzy Logic Operations

These logical operations handle linguistic variables and qualitative reasoning.

• AND (T-Norm): This operation returns the minimum membership value:

• OR (T-Conorm): This operation selects the maximum membership value:

• NOT: This reverses the membership degree:

3. Properties of Fuzzy Relations
   1. Reflexivity

A fuzzy relation R on a set X is reflexive if each element is fully or partially related to itself. The condition for reflexivity is:

Example: Consider a fuzzy relation representing the similarity of pixels in an image. For pixel intensity values x₁, x₂, x₃, the fuzzy relation matrix might look like this:

Here, diagonal elements (μ_R (x₁, x₁) = μ_R (x₂, x₂) = μ_R (x₃, x₃) = 1) show reflexivity since each pixel is fully similar to itself.

1. 2. Symmetry

A fuzzy relation R is symmetric if the membership degree between any two elements is the same in both directions:

Example: In a similarity relation between objects, the membership matrix might be:

Here, μ_R (x₁, x₂) = μ_R (x₂, x₁) = 0.8, showing symmetry.

1. 3. Anti-Symmetry

A fuzzy relation R is anti-symmetric if:

Example: Consider a preference relation where x₁? is preferred over x₂, but the reverse is not true. The membership matrix could be:

Here, μ_R (x₁, x₂) = 0.9, but μ_R (x₂, x₁) = 0, satisfying anti-symmetry.

1. 4. Transitivity

A fuzzy relation R is transitive if the relationship between two elements can be inferred through a third element. The condition is:

Example: Consider a fuzzy relation representing friendship levels between three people. The membership matrix might be:

For x₁?, x₂?, and x₃?, transitivity holds because:

• μ_R? (x₁?, x₃?) = 0.5 and min (μ_R? (x₁?, x₂?), μ_R ?(x₂?, x₃?)) = min (0.6, 0.7) = 0.5.
• Thus, μ_R (x₁, x₃) ≥ min (μ_R (x₁, x₂), μ_R (x₂, x₃)), satisfying transitivity.
  5. Equivalence Relation

A fuzzy relation R is an equivalence relation if it satisfies reflexivity, symmetry, and transitivity. This type of relation groups elements into fuzzy equivalence classes.

Example: In pattern recognition, a fuzzy equivalence relation groups patterns with similar features. For patterns x₁, x₂, x₃?, the membership matrix might look like:

Here, R satisfies reflexivity (μ_R (x, x)=1), symmetry (μ_R(x_i,x_j)=μ_R(x_j,x_i)), and transitivity as shown above.

1. 6. Tolerance Relation

A fuzzy relation is a tolerance relation if it satisfies reflexivity and symmetry, but not necessarily transitivity. This property is useful for approximate similarity.

Example: In linguistic similarity, words like "fast" and "quick" might be related with high membership, but the relationship between "quick" and "rapid" may not necessarily imply a direct relationship between "fast" and "rapid."

Membership Matrix:

Here, reflexivity and symmetry are satisfied, but transitivity may fail as μ_R (x₁, x₃) < min (μ_R (x₁, x₂), μ_R (x₂, x₃)).

1. 7. Fuzzy Ordering

Fuzzy ordering is used to rank elements based on a relation that satisfies anti-symmetry and transitivity.

Example: In decision-making, consider three alternatives x₁, x₂, x₃?. If the relation matrix is:

This satisfies anti-symmetry (μ_R (x₁, x₂) > 0 but μ_R (x₂, x₁) = 0) and transitivity, making it a fuzzy ordering.

RESULTS AND DISCUSSION

Medical images often display tumors that vary in location, type of pathology, shape, size, density, and the extent of the surrounding tissue affected in the region near the tumor (Figure 4). To demonstrate that the proposed methodology possesses strong edge detection capabilities and maintains robustness against moderate noise levels, several studies were conducted.

Figure 4 Experimental Data Set Images

The initial study involved comparing the proposed approach with traditional edge detection methods, including simple gradient operators like Roberts, Prewitt, and Sobel, as well as more advanced techniques such as LoG and Canny. For this comparison, images with minimal noise interference were used. All evaluation metrics during the study were derived using reference images created by a medical expert. Figure 5 illustrates an example of reference edge maps designed by the expert.

Figure 5 (?), (?) Original Images, (b), (d) Contour of Brain Tumor

The metric ranges from 0 to 1, with the maximum value representing the optimal outcome. A value of 1 indicates a perfect match between the images, while a value of 0 signifies no similarity. The percentage of pixels not detected (P_ND) is another key metric, where the ideal value is 0. This metric is computed using the following formula:

The percentage of pixels incorrectly identified as edge pixels, referred to as the percentage of false alarms (P_FA), is calculated as follows:

The Pratt's Figure of Merit (FOM) is another useful measure for assessing the performance of edge detectors. This measure uses the distance between all pairs of points corresponding to quantify the difference between the contours. The FOM, which assesses the similarity between two contours, is defined as:

Here, RE_Cnt represents the number of ideal (reference) edge points, and AE_Cnt represents the number of actual edge points. The variable d_i? is the distance between an edge pixel and the closest edge pixel in the reference, while α is an empirical calibration constant (commonly set to α=1/9, as determined to be optimal by Pratt [15]). The Figure of Merit (FOM) achieves its maximum value of 1 when the images are identical, while a lower value indicates greater dissimilarity. Sensitivity, also known as the true positive rate, measures the percentage of object pixels correctly identified as object pixels. This metric ranges from 0 to 1, with a higher value being ideal. Sensitivity is defined using the following equation:

Accuracy is the proportion of true results. Accuracy gives percentage of how many object and background pixels were exactly detected. The range of metrics lies in between 0 to 1. If accuracy value equals 1, then the output is the same as input. The accuracy is defined as:

Figure 6 Some Results of the Proposed Method