Fractal functions, dimensions and signal analysis / Santo Banerjee, D. Easwaramoorthy, A. Gowrisankar.
Banerjee, Santo, 1976-; Easwaramoorthy, D.; Gowrisankar, A. (Arulprakash)
2021
QA281
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Title Fractal functions, dimensions and signal analysis / Santo Banerjee, D. Easwaramoorthy, A. Gowrisankar.
Author Banerjee, Santo, 1976-
ISBN 9783030626723 (electronic bk.)
3030626725 (electronic bk.)
3030626717
9783030626716
DOI https://doi.org/10.1007/978-3-030-62672-3
Imprint Cham : Springer, 2021.
Language English
Description 1 online resource (140 pages)
Other Standard Identifiers 10.1007/978-3-030-62672-3 doi
Call Number QA281
Dewey Decimal Classification 511/.422
621
Summary This book introduces the fractal interpolation functions (FIFs) in approximation theory to the readers and the concerned researchers in advanced level. FIFs can be used to precisely reconstruct the naturally occurring functions when compared with the classical interpolants. The book focuses on the construction of fractals in metric space through various iterated function systems. It begins by providing the Mathematical background behind the fractal interpolation functions with its graphical representations and then introduces the fractional integral and fractional derivative on fractal functions in various scenarios. Further, the existence of the fractal interpolation function with the countable iterated function system is demonstrated by taking suitable monotone and bounded sequences. It also covers the dimension of fractal functions and investigates the relationship between the fractal dimension and the fractional order of fractal interpolation functions. Moreover, this book explores the idea of fractal interpolation in the reconstruction scheme of illustrative waveforms and discusses the problems of identification of the characterizing parameters. In the application section, this research compendium addresses the signal processing and its Mathematical methodologies. A wavelet-based denoising method for the recovery of electroencephalogram (EEG) signals contaminated by nonstationary noises is presented, and the author investigates the recognition of healthy, epileptic EEG and cardiac ECG signals using multifractal measures. This book is intended for professionals in the field of Mathematics, Physics and Computer Science, helping them broaden their understanding of fractal functions and dimensions, while also providing the illustrative experimental applications for researchers in biomedicine and neuroscience.
Access Note Access limited to authorized users.
Digital File Characteristics text file
PDF
Source of Description Online resource; title from PDF title page (SpringerLink, viewed February 18, 2021).
Added Author Easwaramoorthy, D.
Gowrisankar, A. (Arulprakash)
Series Understanding complex systems.
Available in Other Form Fractal Functions, Dimensions and Signal Analysis.
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Intro
Preface
Contents
1 Mathematical Background of Deterministic Fractals
1.1 Introduction
1.2 Iterated Function System
1.3 Countable Iterated Function System
1.4 Local Countable Iterated Function System
1.4.1 Local Iterated Function System
1.4.2 Existence and Analytical Properties of LCIFS
1.5 Fractal Dimension
1.6 Generalized Fractal Dimensions
1.6.1 Some Special Cases
1.6.2 Limiting Cases of Generalized Fractal Dimensions
1.6.3 Range of Generalized Fractal Dimensions
1.7 Concluding Remarks
2 Fractal Functions
2.1 Introduction

2.2 Interpolation Functions
2.3 Fractal Interpolation Function
2.4 Hidden Variable Fractal Interpolation Function
2.5 Classical Calculus on Fractal Interpolation Functions
2.6 Concluding Remarks
3 Fractional Calculus on Fractal Functions
3.1 Introduction
3.2 Linear Fractal Interpolation Function
3.3 Riemann-Liouville Fractional Calculus Quadratic FIF
3.4 Fractal Dimension
3.5 Fractional Calculus of Quadratic FIF with Variable Scaling Factors
3.6 Concluding Remarks
4 Fractal Interpolation Function for Countable Data
4.1 Introduction

4.2 Existence of FIF for Countable Data Set
4.3 Fractional Calculus on Interpolation Function of Sequence of Data
4.4 Concluding Remarks
5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification
5.1 Introduction
5.2 Experimental Signals
5.2.1 Synthetic Weierstrass Signals
5.2.2 Clinical EEG Signals
5.3 Statistical Methods
5.3.1 ANOVA Test
5.3.2 Box Plot
5.3.3 Normal Probability Plot
5.4 Development of Multifractal Analysis in EEG Signal Classification
5.4.1 Modified Generalized Fractal Dimensions

5.4.2 Improved Generalized Fractal Dimensions
5.4.3 Advanced Generalized Fractal Dimensions
5.4.4 Methods to Analyze the Fractal Time Signals
5.4.5 Results and Discussions
5.5 Multifractal-Wavelet Based Denoising in EEG Signal Classification
5.5.1 Discrete Wavelet Transform
5.5.2 Wavelet Denoising of Signals
5.5.3 Results and Discussions
5.6 Concluding Remarks
6 Fuzzy Multifractal Analysis in ECG Signal Classification
6.1 Introduction
6.2 Fuzzy Multifractal Analysis for Fractal Signals
6.2.1 Fuzzy Renyi Entropy

6.2.2 Fuzzy Generalized Fractal Dimensions for Signals
6.3 Fuzzy Generalized Fractal Dimensions for Deterministic Fractal Waveforms
6.4 Experimental ECG Data
6.5 Fuzzy Generalized Fractal Dimensions for Clinical ECG Signals
6.6 Concluding Remarks
Appendix References

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