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Topical Collection "Wavelets, Fractals and Information Theory"

Editor

Guest Editor
Prof. Dr. Carlo Cattani

1. Engineering School (DEIM), University of Tuscia, Largo dell'Università, 01100 Viterbo, Italy
2. Ton Duc Thang University, HCMC, Vietnam
Website | E-Mail
Phone: +39 3207406560
Interests: wavelets; fractals; fractional calculus; dynamical systems; data analysis; time series analysis; image analysis; computer science; computational methods; composite materials; elasticity; nonlinear waves

Topical Collection Information

Dear Colleagues,

Wavelet analysis and fractals are playing fundamental roles in science, engineering applications, and information theory. Wavelets and fractals are the most suitable methods for the analysis of complex systems, localized phenomena, singular solutions, non-differentiable functions, and, in general, nonlinear problems. Nonlinearity and non-regularity usually characterize the complexity of a problem; they are thus the most-studied features in order to approach a solution to complex problems. Wavelets, fractals, and fractional calculus might also help to improve the analysis of the entropy and complexity of a system.

In information theory, entropy encoding might be considered a sort of compression in a quantization process, and this can be further investigated using wavelet compression. There are many types of definition of entropy that are very useful in the engineering and applied sciences, such as Shannon–Fano entropy, Kolmogorov entropy, etc. However, only entropy encoding is optimal for the complexity of large data analysis, such as in data storage. In fact, the principal advantage of modelling a complex system via wavelet analysis is the minimization of the memory space for storage or transmission. Moreover, this kind of approach reveals some new aspects and promising perspectives for many other kinds of applied and theoretical problems. For instance, in engineering applications, the best way to model traffic in wireless communications is based on fractal geometry, whereas the data are efficiently studied on a wavelet basis.

This Topical Collection will also be an opportunity to extend the research fields of image processing, differential/integral equations, number theory and special functions, image segmentation, the sparse component analysis approach, generalized multiresolution analysis, and entropy as a measure of all aspects of the theoretical and practical studies of mathematics, physics, and engineering.

The main topics of this Topical Collection include (but are not limited to):

  • Entropy encoding, wavelet compression, and information theory
  • Fractals, non-differentiable functions; Theoretical and applied analytical problems of fractal type, fractional equations
  • Fractals, entropy and complexity
  • Fractals, wavelets, fractional methods in the stochastic process, stochastic equations
  • Fractal and wavelet solutions of fractional differential equations
  • Wavelet analysis, integral transforms and applications
  • Wavelets, fractals and fractional methods in fault diagnosis, in signal analysis, in nonlinear time series
  • Wavelet-fractal entropy encoding and computational mathematics in data analysis and time series, including in image analysis
  • Wavelet–fractal approach
  • Fractional nonlinear equations
  • Chaotic dynamics
  • Artifical neural networks.

Prof. Dr. Carlo Cattani
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the collection website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Fractal
  • Fractional
  • Wavelet
  • Entropy
  • Nonlinear time series
  • Data analysis
  • Image Analysis
  • Dynamical System
  • Chaos
  • Differential Operators

Related Special Issues

Published Papers (4 papers)

2019

Jump to: 2018

Open AccessArticle Multimode Decomposition and Wavelet Threshold Denoising of Mold Level Based on Mutual Information Entropy
Entropy 2019, 21(2), 202; https://doi.org/10.3390/e21020202
Received: 1 February 2019 / Revised: 18 February 2019 / Accepted: 19 February 2019 / Published: 21 February 2019
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Abstract
The continuous casting process is a continuous, complex phase transition process. The noise components of the continuous casting process are complex, the model is difficult to establish, and it is difficult to separate the noise and clear signals effectively. Owing to these demerits, [...] Read more.
The continuous casting process is a continuous, complex phase transition process. The noise components of the continuous casting process are complex, the model is difficult to establish, and it is difficult to separate the noise and clear signals effectively. Owing to these demerits, a hybrid algorithm combining Variational Mode Decomposition (VMD) and Wavelet Threshold denoising (WTD) is proposed, which involves multiscale resolution and adaptive features. First of all, the original signal is decomposed into several Intrinsic Mode Functions (IMFs) by Empirical Mode Decomposition (EMD), and the model parameter K of the VMD is obtained by analyzing the EMD results. Then, the original signal is decomposed by VMD based on the number of IMFs K, and the Mutual Information Entropy (MIE) between IMFs is calculated to identify the noise dominant component and the information dominant component. Next, the noise dominant component is denoised by WTD. Finally, the denoised noise dominant component and all information dominant components are reconstructed to obtain the denoised signal. In this paper, a comprehensive comparative analysis of EMD, Ensemble Empirical Mode Decomposition (EEMD), Complementary Empirical Mode Decomposition (CEEMD), EMD-WTD, Empirical Wavelet Transform (EWT), WTD, VMD, and VMD-WTD is carried out, and the denoising performance of the various methods is evaluated from four perspectives. The experimental results show that the hybrid algorithm proposed in this paper has a better denoising effect than traditional methods and can effectively separate noise and clear signals. The proposed denoising algorithm is shown to be able to effectively recognize different cast speeds. Full article
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Open AccessArticle Combining Multi-Scale Wavelet Entropy and Kernelized Classification for Bearing Multi-Fault Diagnosis
Entropy 2019, 21(2), 152; https://doi.org/10.3390/e21020152
Received: 9 January 2019 / Revised: 26 January 2019 / Accepted: 26 January 2019 / Published: 5 February 2019
PDF Full-text (478 KB) | HTML Full-text | XML Full-text
Abstract
Discriminative feature extraction and rolling element bearing failure diagnostics are very important to ensure the reliability of rotating machines. Therefore, in this paper, we propose multi-scale wavelet Shannon entropy as a discriminative fault feature to improve the diagnosis accuracy of bearing fault under [...] Read more.
Discriminative feature extraction and rolling element bearing failure diagnostics are very important to ensure the reliability of rotating machines. Therefore, in this paper, we propose multi-scale wavelet Shannon entropy as a discriminative fault feature to improve the diagnosis accuracy of bearing fault under variable work conditions. To compute the multi-scale wavelet entropy, we consider integrating stationary wavelet packet transform with both dispersion (SWPDE) and permutation (SWPPE) entropies. The multi-scale entropy features extracted by our proposed methods are then passed on to the kernel extreme learning machine (KELM) classifier to diagnose bearing failure types with different severities. In the end, both the SWPDE–KELM and the SWPPE–KELM methods are evaluated on two bearing vibration signal databases. We compare these two feature extraction methods to a recently proposed method called stationary wavelet packet singular value entropy (SWPSVE). Based on our results, we can say that the diagnosis accuracy obtained by the SWPDE–KELM method is slightly better than the SWPPE–KELM method and they both significantly outperform the SWPSVE–KELM method. Full article
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Open AccessArticle Parameter Identification of Fractional-Order Discrete Chaotic Systems
Entropy 2019, 21(1), 27; https://doi.org/10.3390/e21010027
Received: 17 December 2018 / Revised: 27 December 2018 / Accepted: 27 December 2018 / Published: 1 January 2019
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Abstract
Research on fractional-order discrete chaotic systems has grown in recent years, and chaos synchronization of such systems is a new topic. To address the deficiencies of the extant chaos synchronization methods for fractional-order discrete chaotic systems, we proposed an improved particle swarm optimization [...] Read more.
Research on fractional-order discrete chaotic systems has grown in recent years, and chaos synchronization of such systems is a new topic. To address the deficiencies of the extant chaos synchronization methods for fractional-order discrete chaotic systems, we proposed an improved particle swarm optimization algorithm for the parameter identification. Numerical simulations are carried out for the Hénon map, the Cat map, and their fractional-order form, as well as the fractional-order standard iterated map with hidden attractors. The problem of choosing the most appropriate sample size is discussed, and the parameter identification with noise interference is also considered. The experimental results demonstrate that the proposed algorithm has the best performance among the six existing algorithms and that it is effective even with random noise interference. In addition, using two samples offers the most efficient performance for the fractional-order discrete chaotic system, while the integer-order discrete chaotic system only needs one sample. Full article
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2018

Jump to: 2019

Open AccessArticle Dynamics Analysis of a New Fractional-Order Hopfield Neural Network with Delay and Its Generalized Projective Synchronization
Entropy 2019, 21(1), 1; https://doi.org/10.3390/e21010001
Received: 27 November 2018 / Revised: 13 December 2018 / Accepted: 18 December 2018 / Published: 20 December 2018
PDF Full-text (3115 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, a new three-dimensional fractional-order Hopfield-type neural network with delay is proposed. The system has a unique equilibrium point at the origin, which is a saddle point with index two, hence unstable. Intermittent chaos is found in this system. The complex [...] Read more.
In this paper, a new three-dimensional fractional-order Hopfield-type neural network with delay is proposed. The system has a unique equilibrium point at the origin, which is a saddle point with index two, hence unstable. Intermittent chaos is found in this system. The complex dynamics are analyzed both theoretically and numerically, including intermittent chaos, periodicity, and stability. Those phenomena are confirmed by phase portraits, bifurcation diagrams, and the Largest Lyapunov exponent. Furthermore, a synchronization method based on the state observer is proposed to synchronize a class of time-delayed fractional-order Hopfield-type neural networks. Full article
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Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

Tentative title: Fundamental Morphodynamics in Fractal Growth
Authors: J.R. Nicolás-Carlock (1), J.M. Solano-Altamirano (2), and J.L. Carrillo-Estrada (3)
Affiliation: (1) Instituto de Investigaciones Jurídicas, Universidad Nacional Autónoma de México, Ciudad de México, México; (2) Facultad de Ciencias Químicas, Benemérita Universidad Autónoma de Puebla, Puebla, México; (3) Instituto de Física, Benemérita Universidad Autónoma de Puebla, Puebla, México.
Entropy EISSN 1099-4300 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
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