Fractional Calculus in Signal, Imaging Processing and Machine Learning

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: 31 December 2024 | Viewed by 12459

Special Issue Editor


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Guest Editor
College of Computer Science, Sichuan University, Chengdu 610065, China
Interests: application of fractional calculus and fractional partial differential equation to signal analysis; signal processing; image processing; circuits and systems; machine intelligence
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Special Issue Information

Dear Colleagues,

Fractional-order intelligent information processing refers to many fields of science and engineering that incorporate concepts from fractional calculus into their modeling and design. Fractional calculus is a generalization of the classical integer-order differentiation and integration theory. The theory and application of fractional calculus show that the fractional calculus operator is the best description for many complex natural or social phenomena. The fractional order can provide additional freedom of design for various applications. Due to its many unique characteristics, fractional calculus is being widely explored across many fields, such as signal processing, image processing, machine learning, etc. Additionally, fractional calculus is also being explored in computing law, neuromorphic computing, etc.

The focus of this Special Issue is to continue to advance research on topics relating to fractional calculus in signal processing, image processing, and machine learning. Topics that are invited for submission include (but are not limited to):

  • Fractional-order signal processing;
  • Fractional-order image processing;
  • Fractional-order machine learning;
  • Fractional-order computing law;
  • Fractional-order neuromorphic computing.

Prof. Dr. Yifei Pu
Guest Editor

Manuscript Submission Information

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Published Papers (9 papers)

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Research

17 pages, 5814 KiB  
Article
COVID-19 Diagnosis by Extracting New Features from Lung CT Images Using Fractional Fourier Transform
by Ali Nokhostin and Saeid Rashidi
Fractal Fract. 2024, 8(4), 237; https://doi.org/10.3390/fractalfract8040237 - 18 Apr 2024
Viewed by 813
Abstract
COVID-19 is a lung disease caused by a coronavirus family virus. Due to its extraordinary prevalence and associated death rates, it has spread quickly to every country in the world. Thus, achieving peaks and outlines and curing different types of relapses is extremely [...] Read more.
COVID-19 is a lung disease caused by a coronavirus family virus. Due to its extraordinary prevalence and associated death rates, it has spread quickly to every country in the world. Thus, achieving peaks and outlines and curing different types of relapses is extremely important. Given the worldwide prevalence of coronavirus and the participation of physicians in all countries, information has been gathered regarding the properties of the virus, its diverse types, and the means of analyzing it. Numerous approaches have been used to identify this evolving virus. It is generally considered the most accurate and acceptable method of examining the patient’s lungs and chest through a CT scan. As part of the feature extraction process, a method known as fractional Fourier transform (FrFT) has been applied as one of the time-frequency domain transformations. The proposed method was applied to a database consisting of 2481 CT images. Following the transformation of all images into equal sizes and the removal of non-lung areas, multiple combination windows are used to reduce the number of features extracted from the images. In this paper, the results obtained for KNN and SVM classification have been obtained with accuracy values of 99.84% and 99.90%, respectively. Full article
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19 pages, 1872 KiB  
Article
Fractional Fuzzy Neural System: Fractional Differential-Based Compensation Prediction for Reputation Infringement Cases
by Ni Zhang, Wu-Yang Zhu, Peng Jin, Guo Huang and Yi-Fei Pu
Fractal Fract. 2024, 8(3), 172; https://doi.org/10.3390/fractalfract8030172 - 16 Mar 2024
Viewed by 759
Abstract
With the rise of social media and the internet, the rapid dissemination of information has increased the likelihood of reputation infringement. This study utilizes judicial big data and AI to analyze intrinsic connections in reputation infringement cases, aiding judges in delivering consistent rulings. [...] Read more.
With the rise of social media and the internet, the rapid dissemination of information has increased the likelihood of reputation infringement. This study utilizes judicial big data and AI to analyze intrinsic connections in reputation infringement cases, aiding judges in delivering consistent rulings. The challenge lies in balancing freedom of speech with the right to reputation and addressing the ambiguity and subjectivity in infringement cases. This research constructs a structured reputation infringement case dataset from Chinese Judgments Online. It introduces a Fractional Fuzzy Neural System (FFNS) to tackle the vagueness in reputation infringement acts and judicial language, enhancing prediction accuracy for case outcomes. The FFNS, integrating fractional calculus, fuzzy logic, and neural networks, excels in adaptability and nonlinear modeling. It uses fractional order fuzzy membership functions to depict the extent and severity of reputation infringement accurately, combining these outputs with neural networks for predictive analysis. The result is a more precise adjudication tool, demonstrating significant potential for judicial application. Full article
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13 pages, 415 KiB  
Article
Accelerated Gradient Descent Driven by Lévy Perturbations
by Yuquan Chen, Zhenlong Wu, Yixiang Lu, Yangquan Chen and Yong Wang
Fractal Fract. 2024, 8(3), 170; https://doi.org/10.3390/fractalfract8030170 - 14 Mar 2024
Viewed by 825
Abstract
In this paper, we mainly consider two kinds of perturbed accelerated gradient descents driven by Lévy perturbations, which is of great importance for enhancing the global search ability. By using Lévy representation, Lévy perturbations can be divided into two parts: small jumps and [...] Read more.
In this paper, we mainly consider two kinds of perturbed accelerated gradient descents driven by Lévy perturbations, which is of great importance for enhancing the global search ability. By using Lévy representation, Lévy perturbations can be divided into two parts: small jumps and large jumps, whose properties are then carefully discussed. By introducing the concept of attraction domain for local minima, Makovian transition properties are proven for the proposed two perturbed accelerated gradient descents with different infinitesimal matrices. Finally, all the results are extended to the vector case and two simulation examples are provided to validate all the conclusions. Full article
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19 pages, 6142 KiB  
Article
A Bearing Fault Diagnosis Method under Small Sample Conditions Based on the Fractional Order Siamese Deep Residual Shrinkage Network
by Tao Li, Xiaoting Wu, Zhuhui Luo, Yanan Chen, Caichun He, Rongjun Ding, Changfan Zhang and Jun Yang
Fractal Fract. 2024, 8(3), 134; https://doi.org/10.3390/fractalfract8030134 - 26 Feb 2024
Viewed by 971
Abstract
A bearing fault is one of the major causes of rotating machinery faults. However, in real industrial scenarios, the harsh and complex environment makes it very difficult to collect sufficient fault data. Due to this limitation, most of the current methods cannot accurately [...] Read more.
A bearing fault is one of the major causes of rotating machinery faults. However, in real industrial scenarios, the harsh and complex environment makes it very difficult to collect sufficient fault data. Due to this limitation, most of the current methods cannot accurately identify the fault type in cases with limited data, so timely maintenance cannot be conducted. In order to solve this problem, a bearing fault diagnosis method based on the fractional order Siamese deep residual shrinkage network (FO-SDRSN) is proposed in this paper. After data collection, all kinds of vibration data are first converted into two-dimensional time series feature maps, and these feature maps are divided into the same or different types of fault sample pairs. Then, a Siamese network based on the deep residual shrinkage network (DRSN) is used to extract the features of the fault sample pairs, and the fault type is determined according to the features. After that, the contrastive loss function and diagnostic loss function of the sample pairs are combined, and the network parameters are continuously optimized using the fractional order momentum gradient descent method to reduce the loss function. This improves the accuracy of fault diagnosis with a small sample training dataset. Finally, four small sample datasets are used to verify the effectiveness of the proposed method. The results show that the FO-SDRSN method is superior to other advanced methods in terms of training accuracy and stability under small sample conditions. Full article
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19 pages, 6996 KiB  
Article
Research on Application of Fractional Calculus Operator in Image Underlying Processing
by Guo Huang, Hong-ying Qin, Qingli Chen, Zhanzhan Shi, Shan Jiang and Chenying Huang
Fractal Fract. 2024, 8(1), 37; https://doi.org/10.3390/fractalfract8010037 - 5 Jan 2024
Viewed by 1269
Abstract
Fractional calculus extends traditional, integer-based calculus to include non-integer orders, offering a powerful tool for a range of engineering applications, including image processing. This work delves into the utility of fractional calculus in two crucial aspects of image processing: image enhancement and denoising. [...] Read more.
Fractional calculus extends traditional, integer-based calculus to include non-integer orders, offering a powerful tool for a range of engineering applications, including image processing. This work delves into the utility of fractional calculus in two crucial aspects of image processing: image enhancement and denoising. We explore the foundational theories of fractional calculus together with its amplitude–frequency characteristics. Our focus is on the effectiveness of fractional differential operators in enhancing image features and reducing noise. Experimental results reveal that fractional calculus offers unique benefits for image enhancement and denoising. Specifically, fractional-order differential operators outperform their integer-order counterparts in accentuating details such as weak edges and strong textures in images. Moreover, fractional integral operators excel in denoising images, not only improving the signal-to-noise ratio but also better preserving essential features such as edges and textures when compared to traditional denoising techniques. Our empirical results affirm the effectiveness of the fractional-order calculus-based image-processing approach in yielding optimal results for low-level image processing. Full article
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34 pages, 13989 KiB  
Article
Overcoming Nonlinear Dynamics in Diabetic Retinopathy Classification: A Robust AI-Based Model with Chaotic Swarm Intelligence Optimization and Recurrent Long Short-Term Memory
by Yusuf Bahri Özçelik and Aytaç Altan
Fractal Fract. 2023, 7(8), 598; https://doi.org/10.3390/fractalfract7080598 - 3 Aug 2023
Cited by 67 | Viewed by 2246
Abstract
Diabetic retinopathy (DR), which is seen in approximately one-third of diabetes patients worldwide, leads to irreversible vision loss and even blindness if not diagnosed and treated in time. It is vital to limit the progression of DR disease in order to prevent the [...] Read more.
Diabetic retinopathy (DR), which is seen in approximately one-third of diabetes patients worldwide, leads to irreversible vision loss and even blindness if not diagnosed and treated in time. It is vital to limit the progression of DR disease in order to prevent the loss of vision in diabetic patients. It is therefore essential that DR disease is diagnosed at an early phase. Thanks to retinal screening at least twice a year, DR disease can be diagnosed in its early phases. However, due to the variations and complexity of DR, it is really difficult to determine the phase of DR disease in current clinical diagnoses. This paper presents a robust artificial intelligence (AI)-based model that can overcome nonlinear dynamics with low computational complexity and high classification accuracy using fundus images to determine the phase of DR disease. The proposed model consists of four stages, excluding the preprocessing stage. In the preprocessing stage, fractal analysis is performed to reveal the presence of chaos in the dataset consisting of 12,500 color fundus images. In the first stage, two-dimensional stationary wavelet transform (2D-SWT) is applied to the dataset consisting of color fundus images in order to prevent information loss in the images and to reveal their characteristic features. In the second stage, 96 features are extracted by applying statistical- and entropy-based feature functions to approximate, horizontal, vertical, and diagonal matrices of 2D-SWT. In the third stage, the features that keep the classifier performance high are selected by a chaotic-based wrapper approach consisting of the k-nearest neighbor (kNN) and chaotic particle swarm optimization algorithms (CPSO) to cope with both chaoticity and computational complexity in the fundus images. At the last stage, an AI-based classification model is created with the recurrent neural network-long short-term memory (RNN-LSTM) architecture by selecting the lowest number of feature sets that can keep the classification performance high. The performance of the DR disease classification model was tested on 2500 color fundus image data, which included five classes: no DR, mild non-proliferative DR (NPDR), moderate NPDR, severe NPDR, and proliferative DR (PDR). The robustness of the DR disease classification model was confirmed by the 10-fold cross-validation. In addition, the classification performance of the proposed model is compared with the support vector machine (SVM), which is one of the machine learning techniques. The results obtained show that the proposed model can overcome nonlinear dynamics in color fundus images with low computational complexity and is very effective and successful in precisely diagnosing all phases of DR disease. Full article
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27 pages, 448 KiB  
Article
A Light-Ray Approach to Fractional Fourier Optics
by Éric Fogret and Pierre Pellat-Finet
Fractal Fract. 2023, 7(7), 505; https://doi.org/10.3390/fractalfract7070505 - 27 Jun 2023
Viewed by 941
Abstract
A light ray in space is characterized by two vectors: (i) a transverse spatial vector associated with the point where the ray intersects a given spherical cap; (ii) an angular-frequency vector which defines the ray direction of propagation. Given a light ray propagating [...] Read more.
A light ray in space is characterized by two vectors: (i) a transverse spatial vector associated with the point where the ray intersects a given spherical cap; (ii) an angular-frequency vector which defines the ray direction of propagation. Given a light ray propagating from a spherical emitter to a spherical receiver, a linear equation is established that links its representative vectors on the emitter and on the receiver. The link is expressed by means of a matrix which is not homogeneous, since it involves both spatial and angular variables (having distinct physical dimensions). Indeed, the matrix becomes a homogeneous rotation matrix after scaling the previous variables with appropriate dimensional coefficients. When applied to diffraction, in the framework of a scalar theory, the scaling operation results directly in introducing fractional-order Fourier transformations as mathematical expressions of Fresnel diffraction phenomena. Linking angular-frequency vectors and spatial frequencies results in an interpretation of the notion of a spherical angular spectrum. Accordance of both homogeneous and non-homogeneous ray matrices with the Huygens–Fresnel principle is examined. The proposed ray-matrix representation of diffraction is also applied to coherent imaging through a lens. Full article
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27 pages, 15527 KiB  
Article
A Fractional-Order Total Variation Regularization-Based Method for Recovering Geiger-Mode Avalanche Photodiode Light Detection and Ranging Depth Images
by Da Xie, Xinjian Wang, Chunyang Wang, Kai Yuan, Xuyang Wei, Xuelian Liu and Tingsheng Huang
Fractal Fract. 2023, 7(6), 445; https://doi.org/10.3390/fractalfract7060445 - 31 May 2023
Cited by 2 | Viewed by 1021
Abstract
High-quality image restoration is typically challenging due to low signal–to–background ratios (SBRs) and limited statistics frames. To address these challenges, this paper devised a method based on fractional-order total variation (FOTV) regularization for recovering Geiger-mode avalanche photodiode (GM-APD) light detection and [...] Read more.
High-quality image restoration is typically challenging due to low signal–to–background ratios (SBRs) and limited statistics frames. To address these challenges, this paper devised a method based on fractional-order total variation (FOTV) regularization for recovering Geiger-mode avalanche photodiode (GM-APD) light detection and ranging (lidar) depth images. First, the spatial differential peak-picking method was used to extract the target depth image from low SBR and limited frames. FOTV regularization was introduced based on the total variation regularization recovery model, which incorporates the fractional-order differential operator, in order to realize FOTV-regularization-based depth image recovery. These frameworks were used to establish an algorithm for GM-APD depth image recovery based on FOTV. The simulation and experimental results demonstrate that the devised FOTV-recovery algorithm improved the target reduction degree, peak signal–to–noise ratio, and structural similarity index measurement by 76.6%, 3.5%, and 6.9% more than the TV, respectively, in the same SBR and statistic frame conditions. Thus, the devised approach is able to effectively recover GM-APD lidar depth images in low SBR and limited statistic frame conditions. Full article
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23 pages, 7655 KiB  
Article
Image Edge Detection Based on Fractional-Order Ant Colony Algorithm
by Xinyu Liu and Yi-Fei Pu
Fractal Fract. 2023, 7(6), 420; https://doi.org/10.3390/fractalfract7060420 - 23 May 2023
Cited by 1 | Viewed by 1171
Abstract
Edge detection is a highly researched topic in the field of image processing, with numerous methods proposed by previous scholars. Among these, ant colony algorithms have emerged as a promising approach for detecting image edges. These algorithms have demonstrated high efficacy in accurately [...] Read more.
Edge detection is a highly researched topic in the field of image processing, with numerous methods proposed by previous scholars. Among these, ant colony algorithms have emerged as a promising approach for detecting image edges. These algorithms have demonstrated high efficacy in accurately identifying edges within images. For this paper, due to the long-term memory, nonlocality, and weak singularity of fractional calculus, fractional-order ant colony algorithm combined with fractional differential mask and coefficient of variation (FACAFCV) for image edge detection is proposed. If we set the order of the fractional-order ant colony algorithm and fractional differential mask to v=0, the edge detection method we propose becomes an integer-order edge detection method. We conduct experiments on images that are corrupted by multiplicative noise, as well as on an edge detection dataset. Our experimental results demonstrate that our method is able to detect image edges, while also mitigating the impact of multiplicative noise. These results indicate that our method has the potential to be a valuable tool for edge detection in practical applications. Full article
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