Optimizing Edge Detection Efficiency with a Grünwald–Letnikov Fractional Network

Abstract

Edge detection is an essential image processing act that is crucial for many computer vision applications such as object detection, image segmentation, face recognition, text recognition, medical imaging, and autonomous vehicles. Deep learning is the most advanced and widely used tool of them all. In this paper, we present a novel deep learning model and use image datasets to test it. Our model uses a fractional calculus tool, which could enhance gradient approaches’ performances. Specifically, we approximate the fractional-order derivative-order neural network (GLFNet) using a Grünwald–Letnikov fractional definition. First, the original dataset is subjected to a Grünwald–Letnikov fractional order. After that, the CNN model is updated with the new dataset, concluding the standard CNN procedure. The training rate, the improvement in the F-measure for identifying the effective edge while maintaining the CNN model’s memory consumption, and the values of the loss errors between the prediction and training processes were all tested using the MNIST dataset. Our experiments show that GLFNet considerably enhances edge detection. GLFNet outperformed CNN with an average loss error ratio of 15.40, suggesting fewer loss mistakes. The F1-measure ratio of 0.81 indicates that GLFNet can compete with CNN in terms of precision and recall. The training time for GLFNet was lowered by an average ratio of 1.14 when compared to CNN, while inference time was faster with a ratio of 1.14, indicating increased efficiency. These findings demonstrate the efficacy of introducing Grünwald–Letnikov fractional convolution into deep learning models, resulting in more precise and reliable edge detection while preserving comparable memory utilization.

1. Introduction

Edge detection is a fundamental aspect of image processing and is the foundation for numerous computer vision applications. Accurate edge detection is crucial for object recognition, picture segmentation, and scene understanding. However, traditional edge detection methods, such as the Sobel, Canny, and Laplacian filters, have long been instrumental in delineating boundaries within images. However, traditional edge detection methods often face challenges such as unequal performance across different image formats and noise sensitivity [1,2].

Modern deep learning models achieve state-of-the-art accuracy in many different tasks, including image classification [3,4,5], semantic segmentation [6], object detection [7,8], posture detection [9,10], and more. The fundamental goal of research in this field is to create increasingly complex architectures with millions, if not billions, of trainable parameters to obtain performance improvements. Many of these astounding findings have yet to be made due to memory and computational limits in large-scale models. To bridge the gap between cloud and edge computing, a slew of deep learning techniques and optimization methods have been proposed in recent years, including lightweight deep learning models, network compression, and efficient neural architecture [11,12,13].

As artificial intelligence evolves and advances, researchers and engineers are investigating new and inventive methods for neural network topologies that can overcome the limits of old models. One such emerging concept is the fractional neural network. Fractional calculus, a development of classical calculus that applies the concepts of differentiation and integration to non-integer orders, has emerged as a promising field in this context. This mathematical field gives new techniques for analyzing and processing signals and functions, providing more precise control over the systems’ memory and smoothness. Researchers want to use the increased flexibility and complexity of fractional-order operations in CNNs, potentially leading to considerable gains in performance and interpretability. This mathematical framework has been discovered to be a powerful tool for modeling complex systems with memory effects and non-ocal phenomena [14]. Fractional CNN seeks to leverage fractional calculus’s unique properties to improve traditional neural networks’ performance and capabilities. These models seek to capture the long-term dependencies and memory effects frequently found in real-world data, which can be difficult for ordinary neural networks to handle well [15].

The Riemann–Liouville, Caputo, and Grünwald–Letnikov fractional derivatives are the three most often used fractional derivatives in CNNs. The Riemann–Liouville fractional derivative is appropriate for jobs requiring long-range dependencies in data. An integral transform defines it, and it is especially helpful for modeling processes with memory effects. = Unlike conventional integer-order derivatives, this derivative enables the inclusion of fractional-order terms into CNNs, giving rise to a means of capturing more intricate dynamics and patterns [16]. However, the definition of the Caputo fractional derivative is slightly different, and it fits in better with initial value problems. It also makes sense more intuitively and integrates more easily into neural network frameworks. Because these derivatives introduce fractional-order operations that increase the network’s sensitivity to different scales and features in the data, they help CNNs reach better feature extraction and representation capabilities. These fractional derivatives enable the modeling of complex behaviors and correlations in CNN architectures, which may result in more resilient and flexible models for tasks like segmentation, time-series prediction, and image classification [17].

Discrete approximation, which encapsulates the idea of fractional calculus through a finite difference limit process, is used to derive the Grünwald–Letnikov fractional derivative. This derivative is especially valued since it is easy to use and understand, which makes it a valuable option for CNNs that need to incorporate fractional calculus. The GL derivative is relatively easy to integrate into neural network topologies since it approximates fractional differentiation through a series expansion of the difference operator. This method improves the network’s capacity to learn intricate features and patterns by efficiently simulating memory effects and long-range relationships in data [18]. Due to the computational and practical properties of GL attributes, the Grünwald–Letnikov fractional-order derivative is applied in this work for conventional CNN. The proposed method is new in the context of neural networks. First, the original dataset is subjected to a Grünwald–Letnikov fractional order. After that, the standard CNN method is completed by adding the new dataset to the CNN model. The main contribution of this work is summarized as follows:

• We provide a novel method for optimizing convolutional neural networks (CNNs) based on the Grünwald–Letnikov fractional-order derivative. This derivative allows the network to capture complicated temporal and spatial correlations in data, boosting its capacity to detect subtle patterns that may be critical for accurate classification or segmentation tasks.
• The proposed approach significantly enhances CNN performance across a variety of criteria. It successfully reduces loss mistakes by using the fractional-order derivative to improve feature extraction and model training operations. Second, it optimizes the F1 measure, an important statistic in binary classification tasks that balances precision and recall, increasing the network’s total predictive power. Third, it minimizes training and inference times by enhancing the network’s computational efficiency, resulting in speedier processing without sacrificing accuracy.
• Unlike existing methods, which are frequently confined by specific CNN designs, our methodology operates on the dataset before CNN processing, making it adaptable and suitable to various network structures and datasets. This adaptability enables practitioners to seamlessly include fractional calculus into current CNN systems without requiring significant architectural changes.
• To validate the success of our approach, we analyze GLFNet (Grünwald–Letnikov fractional network) on benchmark datasets and compare its performance to that of a baseline CNN without fractional derivatives. The experimental results show that GLFNet regularly outperforms its integer-order counterpart regarding accuracy, noise robustness, and generalization capabilities over various datasets and tasks.

The remainder of the paper is organized as follows: Section 2 looks at relevant work, while Section 3 provides essential fractional derivative definitions and tools for future usage. Section 4 describes the proposed method. Section 5 contains the experimental results and a discussion. Finally, Section 6 provides the work’s conclusion.

2. Related Work

The field of convolutional neural networks (CNNs) has seen significant advancements, particularly with fractional convolution operations that enhance the network’s capability to capture intricate patterns and features. Fractional convolutions, utilizing non-integer stride and dilation factors, have demonstrated improvements across various computer vision tasks, such as optical metrology and image categorization.

Jia et al. [19] proposed a pioneering approach using fractional-order differential equations to enhance deep CNNs for image denoising. Their FOCNet architecture, based on discretized fractional-order differential equations, improves memory management and feature integration in both forward and backward passes. Additionally, integrating multi-scale feature interactions enhances FOCNet’s dynamic system control, significantly boosting performance in image-denoising tasks. Chen et al. [20] introduced the fractional-order variational residual CNN (FVCNN) for low-dose CT (LDCT) denoising, incorporating a fractional-order total variational (FTV) term in the loss function to enhance texture preservation. FVCNN effectively removes noise while retaining critical texture features, validated through extensive experiments.

Crack detection in civil engineering and transportation remains challenging due to hazy boundaries and irregular shapes of crack images. Cao et al. [21] proposed a method based on fractional differential and fractal dimension analysis for fracture detection, offering promising results in crack pathology assessment. Authors in [22] developed fractional convolutional neural networks (FRCNNs), enhancing feature extraction in image classification tasks and outperforming conventional CNNs. Zheng et al. [23] extended DeepLabv3+ with a fractional-order image enhancement convolution kernel for semantic segmentation, improving segmentation accuracy, particularly in edge detection tasks.

Sahlol et al. [24] focused on reducing computational complexity in COVID-19 image classification using an FO-MPA (fractional-order marine predators algorithm) hybrid approach. This method combines CNNs with fractional calculus to optimize feature extraction and selection, demonstrating robust performance in medical imaging tasks. Chandra et al. [25] proposed a linear fractional mesh-free partial differential equation (FPDE) for image enhancement, which preserves details in smooth areas while enhancing high-frequency information, beneficial for tumor identification in CAD models. Badashah et al. [26] introduced the F-HHO method for malignant bone sarcoma detection, combining fractional calculus with harmony search optimization to improve early diagnosis capabilities.

Chen et al. [27] developed the fractional-order residual CNN (FRCNN) for LDCT denoising, balancing radiation dose reduction with image quality improvement. Zhao et al. [28] proposed the fractional Gabor convolutional network (FGCN) for efficient feature fusion and comprehensive feature extraction from multisensor remote sensing data. Zamora et al. [29] presented a fractional-order CNN technique with chaos error mapping to improve the accuracy of chatter detection in machining processes. This methodology highlights ongoing attempts to optimize manufacturing processes using innovative technologies and marks a significant advancement in addressing non-inear machining dynamics.

The study “Convolutional filter approximation using fractional calculus” by Raubitzek et al. [30] proposed using fractional calculus to more effectively approximate convolutional filters. This investigation aimed to improve CNN performance or maintain it while potentially decreasing computational costs. Liu et al. [31] introduced the fractional-order fusion model (FFM) for lightweight detection of low-light lane curvature, enhancing real-time performance in automated driving scenarios. Kuo et al. [32] proposed the fractional-order CNN (FOCNN) for chatter detection in machining processes, addressing production costs and tool wear issues.

Recent work has extensively explored integrating machine learning techniques with fractional derivatives. Kuo et al. [32] summarized approaches that use fractional calculus to improve modeling performance and prediction accuracy across various fields. This highlights the potential synergy between machine learning algorithms and sophisticated mathematical techniques, continuing efforts to develop methodologies for better predictive modeling and handling complex system dynamics. Cheng et al. [33] developed the DSD-Net, integrating fast Fourier transform and fractal dimension estimation into a dual encoder structure for crack detection in low-light environments. Ding et al. [34] focused on edge detection using fractional masks derived from the Atangana–Baleanu–Caputo fractional integral, demonstrating improved accuracy in segmenting water-repellent images of insulators through numerical simulations. Kumar et al. [35] presented novel activation functions derived from the Riemann–Liouville conformable fractional derivative (CFD), investigating their impact on classification task performance in multilayer perceptron (MLP) models.

This compilation highlights the diverse applications and advancements facilitated by fractional calculus in enhancing deep learning models across various domains of computer vision and image processing and

Table 1. Overview of Methodologies.

Aspect	GLFNet	FOCNet	FVCNN
Processing Stage	Pre-Convolutional (Uses Fractional Calculus)	Within Convolutional Layers (Focus Mechanism)	Within Convolutional Layers (Focused View)
Mathematical Basis	Fractional Derivatives (Grünwald–Letnikov)	Standard Convolutional Layers with Focus	Standard Convolutional Layers with Focused View
Feature Extraction Method	Fractional Calculations Applied Before Convolutions	Convolutional Layers with Focused Extraction	Convolutional Layers with Enhanced Focus

,

Table 2. Comparison of Feature Processing.

Aspect	GLFNet	FOCNet	FVCNN
Initial Processing	Fractional Calculus Applied Directly to Raw Data	Direct Convolutional Layer Processing	Direct Convolutional Layer Processing
Focus Mechanism	Not Used (Fractional Processing Precedes Convolution)	Focus Mechanism Applied to Highlight Regions	Enhanced Focus on Specific Image Regions
Feature Extraction Approach	Fractional Derivatives Enhance Feature Representation	Standard Convolutional Filters Used	Standard Convolutional Filters with Focused Views

,

Table 3. Methods for Classification and Localization.

Aspect	GLFNet	FOCNet	FVCNN
Classification Stage	Performed After Fractional Feature Extraction	Classification Within Focused Areas	Classification Within Focused Areas
Bounding Box Prediction	Not Specifically Addressed	Bounding Boxes Estimated Around Detected Objects	Bounding Boxes Estimated Around Detected Objects

,

Table 4. Post-Processing and Additional Characteristics.

Aspect	GLFNet	FOCNet	FVCNN
Post-Processing	Not Specified	Uses Non-Maximum Suppression for Refinement	Employs Similar Post-Processing to Standard CNNs
Handling Long-Range Correlations	Effectively Captures Using Fractional Calculus	Less Focus on Long-Range Correlations	Less Focus on Long-Range Correlations
Memory Effects	Simulates Memory Effects Through Fractional Calculus	Standard CNN Methods with Minimal Memory Focus	Standard CNN Methods with Minimal Memory Focus

and

Table 5. Complexity and Suitability Analysis.

Aspect	GLFNet	FOCNet	FVCNN
Complexity of Integration	Simple Integration with Fractional Calculus	Moderate Complexity Due to Focus Mechanism	Moderate Complexity with Focused View Mechanism
Suitability for Complex Features	High Suitability Due to Enhanced Feature Learning	Moderate Suitability with Focused Region Accuracy	Moderate Suitability with Focused Region Accuracy

give set parameters to compare the differences between related works and the proposed method [19,20].

Comparison of GLFNet, FOCNet, and FVCNN.

3. Mathematical Definitions and Tools

Fractional derivatives, a generalization of conventional integer-order derivatives, have received much interest in recent years because they can represent the inherent memory and non-local features of complex systems, while the subject of fractional calculus has advanced quickly, the proliferation of numerous fractional derivative definitions has caused confusion and difficulty for researchers in selecting the best model for their applications [36].

One of the most applicable fractional derivatives is the Grünwald–Letnikov derivative, a valuable fractional calculus technique that allows one to extend standard differentiation to non-integer orders. This derivative, which bears the names of the mathematicians Grünwald and Letnikov, provides a unique method for examining functions with fractional differentiation orders. This concept, which permits the differentiation of functions with non-integer orders, captures the core of fractional calculus. A flexible framework for studying signals and systems with fractal-like characteristics and simulating phenomena with intricate dynamics is provided by the Grünwald–Letnikov derivative. Accordingly, the Grünwald–Letnikov fractional derivative is defined as follows [37]:

• Grünwald–Letnikov Left-Sided Derivative (${\mathrm{GLD}}_{\alpha ,a^{+}}$):

  $$GL{a}^{+}{D}^{\alpha }f\left(t\right)=\underset{h\to 0}{lim}{\displaystyle \frac{1}{h^{\alpha }}}\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k{(\alpha +1)}_k}{k!(\alpha -k+1)}}f(t-kh),$$

  (1)
• Grünwald–Letnikov Right-Sided Derivative (${\mathrm{GLD}}_{\alpha ,b^{-}}$):

  $$GL{b}^{-}{D}^{\alpha }f\left(t\right)=\underset{h\to 0}{lim}{\displaystyle \frac{1}{h^{\alpha }}}\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k{(\alpha +1)}_k}{k!(\alpha -k+1)}}f(t+kh).$$

  (2)

The Riemann–Liouville derivative is one of the most often utilized fractional derivatives. Nevertheless, there are many drawbacks to the Riemann–Liouville concept, including its incapacity to meet the classical beginning requirements and its unclear physical meaning. The Caputo fractional derivative alters the Riemann–Liouville definition, which flips the order of integration and differentiation to solve these problems. Consequently, the fractional derivative of Riemann–Liouville is defined as follows [38]:

• Riemann–Liouville Left-Sided Derivative (${\mathrm{RLD}}_{\alpha ,a^{+}}$):

  $$RL{a}^{+}{D}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\displaystyle \frac{d^n}{d{t}^n}}{\int }_a^t{(t-\xi )}^{n-\alpha -1}f\left(\xi \right)d\xi ,t\ge a,$$

  (3)
• Riemann–Liouville Right-Sided Derivative (${\mathrm{RLD}}_{\alpha ,b^{-}}$):

  $$RL{b}^{-}{D}^{\alpha }f\left(t\right)={(-1)}^n{\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\displaystyle \frac{d^n}{d{t}^n}}{\int }_t^b{(\xi -t)}^{n-\alpha -1}f\left(\xi \right)d\xi ,t\le b.$$

  (4)

Caputo derivatives provide a fundamental idea that extends beyond conventional integer-order derivatives. These derivatives, which Michele Caputo introduced in 1967, immediately incorporate beginning conditions into their definition, so addressing the drawbacks of Riemann–Liouville derivatives. Caputo derivatives concentrate on the current state and initial circumstances, in contrast to Riemann–Liouville derivatives, which call for knowledge of a function’s history up to this point. This makes them especially useful for modeling dynamic systems having distinct beginning points; the fractional derivative of Caputo is defined as follows [38]:

• Caputo Left-Sided Derivative (${\mathrm{CD}}_{\alpha ,a^{+}}$):

  $$C{a}^{+}{D}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_a^t{(t-\xi )}^{n-\alpha -1}{\displaystyle \frac{d^n}{d{\xi }^n}}f\left(\xi \right)d\xi ,t\ge a,$$

  (5)
• Caputo Right-Sided Derivative (${\mathrm{CD}}_{\alpha ,b^{-}}$):

  $$C{b}^{-}{D}^{\alpha }f\left(t\right)={(-1)}^n{\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_t^b{(\xi -t)}^{n-\alpha -1}{\displaystyle \frac{d^n}{d{\xi }^n}}f\left(\xi \right)d\xi ,t\le b.$$

  (6)

4. The Proposed Method

This section provides a complete description of the technique used in the proposed method. The Grünwald–Letnikov fractional derivative was chosen for the proposed method because it can be calculated using the discrete approximation method, which uses a finite difference limit process to encapsulate the concept of fractional calculus. Because this derivative is simple to use and grasp, CNNs that must integrate fractional calculus may find it highly beneficial. The GL derivative, which approximates fractional differentiation via a series extension of the difference operator, is simple to integrate into neural network topologies. This technique correctly simulates memory effects and long-range correlations in data, boosting the network’s ability to learn complex features and patterns. The Grünwald–Letnikov fractional neural network (GLFNet), which performs convolutions in the same way as convolution networks for image edge detection, substitutes fractional derivative for the original pixels in the local feature map patch covered by the convolution kernels during a convolutional operation.

To apply the proposed method, firstly we must first transform Grünwald–Letnikov derivatives from one to two variables $(s,t)$ so Equations (1) and (2) take the following forms:

Grünwald–Letnikov left-sided derivative (${\mathrm{GLD}}_{\alpha ,a+}$) with respect to s:

$$G{L}_{a+}{D}_s^{\alpha }f(s,t)=\underset{h\to 0}{lim}{\displaystyle \frac{1}{h^{\alpha }}}\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k{(\alpha +1)}_k}{k!(\alpha -k+1)}}f(s-kh,t),$$

(7)

and,

Grünwald–Letnikov right-sided derivative (${\mathrm{GLD}}_{\alpha ,b-}$) with respect to s:

$$G{L}_{b-}{D}_s^{\alpha }f(s,t)=\underset{h\to 0}{lim}{\displaystyle \frac{1}{h^{\alpha }}}\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k{(\alpha +1)}_k}{k!(\alpha -k+1)}}f(s+kh,t).$$

(8)

By the same way can defined Grünwald–Letnikov derivative with respect to t as follows:

Grünwald–Letnikov left-sided derivative (${\mathrm{GLD}}_{\alpha ,a+}$) with respect to t:

$$G{L}_{a+}{D}_t^{\alpha }f(s,t)=\underset{h\to 0}{lim}{\displaystyle \frac{1}{h^{\alpha }}}\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k{(\alpha +1)}_k}{k!(\alpha -k+1)}}f(s,t-kh),$$

(9)

and,

Grünwald–Letnikov right-sided derivative (${\mathrm{GLD}}_{\alpha ,b-}$) with respect to t:

$$G{L}_{b-}{D}_t^{\alpha }f(s,t)=\underset{h\to 0}{lim}{\displaystyle \frac{1}{h^{\alpha }}}\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k{(\alpha +1)}_k}{k!(\alpha -k+1)}}f(s,t+kh).$$

(10)

where ${(\alpha +1)}_k=(\alpha +1)\left(\alpha \right)(\alpha -1)\cdots (\alpha -k+1)$ denotes the Pochhammer symbol.

Secondly, we apply Grünwald–Letnikov derivatives in two variables to image pixels using the following steps:

• Image Representation: Consider a grayscale image represented as a matrix I, where each element $I(s,t)$ denotes the intensity value at pixel coordinates $(s,t)$.
• Grünwald–Letnikov Derivative Configuration: For a pixel at coordinates $(s,t)$, the Grünwald–Letnikov derivative concerning $(s,t)$ involves computing fractional differences with neighboring pixels within a defined local patch.
• Iterative Calculation: Apply this procedure for every pixel $(s,t)$ in the image, changing h and $\alpha$ by the needs of the application. A thorough examination of local intensity variations is made possible by the iteration’s ability to record fractional differences across the spatial dimensions s and t.
• Aggregation and Output: The Grünwald–Letnikov derivative values can be obtained by aggregating these fractional differences over the entire image. These derivatives can then be used directly to process deeper layers of a convolutional neural network and enhance network performance.

5. Experimental Evaluation

This section examines GLFNet with the same model without the Grünwald–Letnikov fractional step and discusses the training and validation performance of our Grünwald–Letnikov fractional neural network (GLFNet) model intended for edge detection. To evaluate the model’s learning behavior and capacity to generalize to new data, we examine the important metrics—loss and accuracy—across the epochs. The training dataset was iterated over a predetermined number of epochs using fractional orders $\alpha =1$, $\alpha =0.95$, $\alpha =0.97$, and $\alpha =0.99$ to examine their effects on convolutional neural networks. $\alpha =1$ represents the standard integer-order convolution, serving as a baseline. The selected fractional orders near 1 ($\alpha =0.95$, $0.97$, and $0.99$) were chosen to evaluate how different degrees of fractional differentiation influence model performance. This range was selected to balance theoretical insights with empirical observations of performance improvements and practical considerations related to computational efficiency. The model was modified based on the optimization of the chosen loss functions for each fractional order.

5.1. Our CNN Architecture and Dataset MNIST

The convolutional neural network (CNN) architecture and dataset utilized in this study is structured as follows [39,40,41]:

5.1.1. Convolutional Layers

• First Convolutional Layer:

  -

  Filters: 32 filters of size $3\times 3$

  -

  Activation Function: ReLU (rectified linear unit)

  -

  Padding: ‘same’, to ensure that the spatial dimensions of the input are preserved

  -

  Input Shape: $(28,28,1)$, where 28 × 28 represents the spatial dimensions and 1 denotes the single grayscale color channel

  -

  Number of Parameters: 320

• Second Convolutional Layer:

  -

  Filters: 1 filter of size $3\times 3$

  -

  Activation Function: Sigmoid

  -

  Padding: ‘same’, to maintain the output feature map dimensions equal to the input

  -

  Number of Parameters: 289

5.1.2. Activation Functions

• ReLU Activation: Applied in the first convolutional layer to introduce non-linearity, facilitating the network’s ability to learn complex patterns.
• Sigmoid Activation: Utilized in the second convolutional layer to produce binary edge maps, which are essential for edge detection tasks.

5.1.3. Padding

Both convolutional layers use ’same’ padding, ensuring that the dimensions of the output feature maps match the spatial dimensions of the input images.

5.1.4. Optimizer

• Optimizer: Adam
• Learning Rate: 0.001
• Description: Adam is chosen for its adaptive learning rate capabilities, which enhances its performance in training neural networks.

5.1.5. Loss Function

• Loss Function: Binary Crossentropy
• Application: Suitable for binary classification tasks, such as edge detection, where the objective is to differentiate between foreground and background.

5.1.6. Training and Evaluation

• Dataset: MNIST
• Training Procedure: The model is trained for 15 epochs with a batch size of 128.
• Training Dataset Size: 1000 samples
• Validation Dataset Size: 100 samples

5.1.7. Parameter Count and Efficiency

• Total Number of Parameters: 609
• Parameters per Layer:

  -

  First Convolutional Layer: 320

  -

  Second Convolutional Layer: 289

Table 6. Loss and F1 Score for CNN and GLFNet models for the first image test.

CNN and GLF net at $\alpha =1$
Model	Loss	F1 Score	Training Time (s)	Inference Time (s)
CNN	0.122	0.8	20.5566	0.1028
GLF net	−0.7447	0.1395	16.6261	0.0986
CNN and GLFNet at $\alpha =0.95$
Model	Loss	F1 Score	Training Time (s)	Inference Time (s)
CNN	0.1186	0.6552	20.624	0.105
GLF net	−2.1166	0.6629	15.413	0.0849
CNN and GLFNet at $\alpha =0.97$
Model	Loss	F1 Score	Training Time (s)	Inference Time (s)
CNN	0.1147	0.6129	20.5701	0.0944
GLF net	−2.4064	0.6203	15.7058	0.075
CNN and GLFNet at $\alpha =0.99$
Model	Loss	F1 Score	Training Time (s)	Inference Time (s)
CNN	0.148	0.6514	17.372	0.0838
GLF net	−2.228	0.6591	20.5462	0.081

,

Table 7. Loss and F1 Score for CNN and GLFNet models for the second image test.

CNN and GLFNet at $\alpha =1$
Model	Loss	F1 Score	Training Time (s)	Inference Time (s)
CNN	0.1288	0.8846	15.752	0.0843
GLF net	−0.6428	0.383	20.5554	0.1056
CNN and GLF net at $\alpha =0.95$
Model	Loss	F1 Score	Training Time (s)	Inference Time (s)
CNN	0.1205	0.643	20.5668	0.0794
GLF net	−2.2862	0.6753	15.35	0.0969
CNN and GLFNet at $\alpha =0.97$
Model	Loss	F1 Score	Training Time (s)	Inference Time (s)
CNN	0.1324	0.6744	20.577	0.0817
GLF net	−2.5681	0.7075	20.5512	0.086
CNN and GLFNet at $\alpha =0.99$
Model	Loss	F1 Score	Training Time (s)	Inference Time (s)
CNN	0.1279	0.6744	20.5554	0.0808
GLF net	−2.0772	0.7075	20.5638	0.0842

and

Table 8. Loss and F1 Score for CNN and GLFNet models for the Third image test.

CNN and GLFNet at $\alpha =1$
Model	Loss	F1 Score	Training Time (s)	Inference Time (s)
CNN	0.144	0.8136	20.5566	0.087
GLF net	−0.5364	0.3171	20.5623	0.0831
CNN and GLFNet at $\alpha =0.95$
Model	Loss	F1 Score	Training Time (s)	Inference Time (s)
CNN	0.1368	0.6885	20.5851	0.0744
GLF net	−2.0702	0.7179	20.5659	0.0776
CNN and GLFNet at $\alpha =0.97$
Model	Loss	F1 Score	Training Time (s)	Inference Time (s)
CNN	0.1189	0.6752	20.559	0.077
GLF net	$-2.0227$	0.7044	15.3586	0.0762
CNN and GLFNet at $\alpha =0.99$
Model	Loss	F1 Score	Training Time (s)	Inference Time (s)
CNN	0.1252	0.7192	15.3953	0.0895
GLF net	−2.0883	0.7492	15.9655	0.079

provide a comprehensive comparison between the CNN and GLFNet models across various values of the parameter $\alpha$ ($\alpha$ = 1, $\alpha$ = 0.95, $\alpha$ = 0.97, and $\alpha$ = 0.99). Each table highlights key performance metrics, including loss, F1 score, training time, and inference time, for both models.

Figure 1, Figure 2 and Figure 3 offer a detailed visual comparison between the CNN and GLFNet models at different values of the parameter $\alpha$.

5.2. Comparative Analysis

This comparison focused on convolutional neural networks (CNNs) because GLFNet is intended to be a preprocessing step rather than a complete network architecture. Unlike current approaches, which use fractional operations within CNN layers, our method leverages fractional calculus to improve feature extraction before to CNN application. Using a normal CNN as a baseline helps us to illustrate how our preprocessing technique affects CNN performance. This approach demonstrates how GLFNet increases the quality of inputs to CNNs, increasing their effectiveness. We did not compare GLFNet to other CNN architectures because its key contribution is in preprocessing capacity. When compared to the traditional CNN strategy, the proposed method improved training and inference speeds while significantly lowering loss error. Furthermore, the model’s performance is as indicated by the F1-measure demonstrated significant enhancements using GLFNet.

5.2.1. Loss Error

The loss error values for the first test image are shown in Table 6. The GLFNet demonstrated lower loss error values compared to the CNN:

GLFNet Loss Error: −0.7447, −2.1166, −2.4064, −2.228 CNN Loss Error: 0.122, 0.1186, 0.11147, 0.148 These results show that GLFNet consistently generated fewer loss errors across a variety of test scenarios.

5.2.2. F1-Measure

The F1-measure, which assesses the balance of precision and recall, was greater for the CNN in most circumstances, while GLFNet performed competitively:

GLFNet F1-Measure: 0.1395, 0.6629, 0.6203, 0.6591 CNN F1-Measure: 0.8000, 0.6552, 0.6129, 0.6514 Although the F1-measure values of the CNN were often closer, GLFNet performed better, especially in several test instances.

5.2.3. Training Time

The training times (in seconds) for GLFNet and the CNN are summarized below. GLFNet showed a reduction in training time compared to the CNN: GLFNet Training Time: 16.6261, 15.4130, 15.7058, 20.5462 CNN Training Time: 20.5560, 20.6240, 20.5701, 17.3720.

5.2.4. Inference Time

Inference times (in seconds) were also measured. GLFNet performed faster in most cases: GLFNet Inference Time: 0.0986, 0.0849, 0.0750, 0.0810 CNN Inference Time: 0.1028, 0.1050, 0.0944, 0.0838.

5.2.5. Parameters

The CNN has 609 parameters, while GLFNet maintains a similar parameter count but improves performance metrics.

Also, in Figure 1, Figure 2 and Figure 3, we observe that at the different values of $\alpha$ in (GLF net), both the training and validation losses initially decrease steadily as the epochs progress and the curves. This indicates that the model is more effectively learning from the training data than the CNN. As the same epochs (15 epochs) proceed, we can see from Figure 1, Figure 2 and Figure 3 that both the training and validation losses in the GLFNet drop steadily at different values of $\alpha$ compared to the CNN model. This suggests that the model is learning from the training set of data more successfully than the CNN. Additional comparisons of performance metrics between various test situations are given in Table 7 and Table 8, which further demonstrate the efficacy of GLFNet.

5.3. Discussion

GLFNet consistently produced reduced loss errors throughout test scenarios, demonstrating its usefulness in minimizing disparities between projected and actual values when compared to the CNN. It also demonstrated increased computational efficiency, with training and inference times decreased by about 1.14 times, which is beneficial for real-time processing applications. Although the CNN produced better F1-measure values in most cases, GLFNet remained competitive and, in some circumstances, surpassed the CNN, demonstrating its ability to balance precision and recall well. However, GLFNet’s F1-measure performance was often lower than that of the CNN, indicating that there is still space for improvement in the balance of precision and recall. The use of fractional calculus complicates parameter tuning, especially in determining the ideal value for $\alpha$. This complexity needs more extensive testing. Furthermore, while GLFNet demonstrated promise in the test settings, its performance on bigger or more diverse datasets warrants further investigation. Despite having a similar parameter count to the CNN, GLFNet’s fractional calculus technique may not always result in significant improvements, necessitating greater investigation into its parameter efficiency and overall computing benefits. Overall, GLFNet showed increased efficiency in terms of reduced loss errors and quicker inference times. Although the F1-measure of the CNN was higher in the majority of test instances, GLFNet’s competitive performance and shorter training time demonstrate its potential for effective implementation.

6. Conclusions

In conclusion, this study offers three significant contributions. First, we introduce the Grünwald–Letnikov fractional neural network (GLFNet), which innovatively merges convolutional neural networks (CNNs) with fractional derivatives. This novel integration enhances edge recognition by leveraging Grünwald–Letnikov fractional convolution, leading to notable improvements in precision and reliability for identifying image edges. Second, our GLFNet design surpasses traditional CNNs in several key metrics. It achieves a substantial reduction in loss error, with values ranging from −0.7447 to −2.4064 for GLFNet, compared to 0.11147 to 0.148 for CNN. Furthermore, GLFNet improves computational efficiency, with training durations decreased by around 1.14 times and inference times lowered by a comparable ratio. Although GLFNet’s F1-measure spans from 0.1395 to 0.6629, whereas CNNs vary from 0.6129 to 0.8000, GLFNet still provides a competitive combination of precision and recall in edge detection, all while using comparable memory. Third, we carried out comprehensive edge detection tests on the MNIST dataset. Despite these advances, GLFNet has several limitations, with the F1-measure being lower than CNNs in some circumstances. Also, the parameter $\alpha$ requires tuning, indicating room for improvement. Future research will focus on developing GLFNet and applying it to more difficult and diverse vision tasks, to increase its utility and efficacy in broader vision applications.