Wavelet Transforms: Kith and Kin serves as an introduction to contemporary aspects of time-frequency analysis encompassing the theories of Fourier transforms, wavelet transforms and their respective offshoots.
This book is the first of its kind totally devoted to the treatment of continuous signals and it systematically encompasses the theory of Fourier transforms, wavelet transforms, geometrical wavelet transforms and their ramifications. The authors intend to motivate and stimulate interest among mathematicians, computer scientists, engineers and physical, chemical and biological scientists.
The text is written from the ground up with target readers being senior undergraduate and first-year graduate students and it can serve as a reference for professionals in mathematics, engineering and applied sciences.
Features
- Flexibility in the book’s organization enables instructors to select chapters appropriate to courses of different lengths, emphasis and levels of difficulty
- Self-contained, the text provides an impetus to the contemporary developments in the signal processing aspects of wavelet theory at the forefront of research
- A large number of worked-out examples are included
- Every major concept is presented with explanations, limitations and subsequent developments, with emphasis on applications in science and engineering
- A wide range of exercises are incoporated in varying levels from elementary to challenging so readers may develop both manipulative skills in theory wavelets and deeper insight
- Answers and hints for selected exercises appear at the end
The origin of the theory of wavelet transforms dates back to the 1980s as an outcome of the intriguing efforts of mathematicians, physicists and engineers. Owing to the lucid mathematical framework and versatile applicability, the theory of wavelet transforms is now a nucleus of shared aspirations and ideas.
List of Figures xix
1 The Fourier Transforms 1
1.1 Introduction
1.2 The Fourier Transform
1.2.1 Definition and Examples
1.2.2 Basic Properties of the Fourier Transform
1.2.3 Convolution and Correlation
1.2.4 Shannon’s Sampling Theorem
1.2.5 Uncertainty Principle for the Fourier Transform
1.3 The Fractional Fourier Transform
1.3.1 Definition and Basic Properties
1.3.2 Fractional Convolution and Correlation
1.3.3 Uncertainty Principle for the Fractional Fourier Transform
1.4 The Linear Canonical Transform
1.4.1 Definition and Basic Properties
1.4.2 Linear Canonical Convolution and Correlation
1.4.3 Applications of Linear Canonical Transform to Differential Equations
1.4.4 Uncertainty Principle for the Linear Canonical Transform
1.5 The Special Affine Fourier Transform
1.5.1 Definition and Basic Properties
1.5.2 Special Affine Convolution and Correlation
1.6 The Quadratic-phase Fourier Transform
1.6.1 Definition and Basic Properties
1.6.2 Quadratic-phase Convolution and Correlation
1.6.3 Applications of Quadratic-phase Fourier Transform
1.7 The Two-dimensional Fourier Transform
1.8 Exercises
2 The Windowed Fourier Transforms 119
2.1 Introduction
2.2 The Windowed Fourier Transform
2.2.1 Analysis of Window Functions
2.2.2 Definition and Basic Properties
2.2.3 Time-Frequency Resolution and Uncertainty Principles
2.3 The Windowed Fractional Fourier Transform
2.3.1 Definition and Basic Properties
2.3.2 Time-Fractional-Frequency Resolution
2.4 The Windowed Linear Canonical Transform
2.4.1 Definition and Basic Properties
2.4.2 Resolution of Windowed Linear Canonical Transform and Uncertainty
Principle
2.5 The Windowed Special Affine Fourier Transform
2.6 The Windowed Quadratic-Phase Fourier Transform
2.7 The Directional Windowed Fourier Transform
2.8 Exercises
3 The Wavelet Transforms and Kin 193
3.1 Introduction
3.2 The Wavelet Transform
3.2.1 Wavelet Transform: A Group Theoretical Approach
3.2.2 Wavelet Transform: An Analytical Perspective
3.3 The Stockwell Transform
3.3.1 Definition and Basic Properties
3.4 The Two-dimensional Wavelet Transform
3.4.1 Two-dimensional Wavelet Transform: A Group Theoretic Approach
3.4.2 Two-dimensional Wavelet Transform: An Analytical Perspective
3.4.3 Different Classes of Two-dimensional Wavelets
3.4.4 Representation Classes and Energy Densities for Two-dimensional
Wavelet Transform
3.5 The Ridgelet Transform
3.5.1 Radon Transform
3.5.2 Ridgelet Transform: Definition and Basic Properties
3.6 The Curvelet and Ripplet Transforms
3.6.1 Curvelet Transform
3.6.2 Ripplet Transform
3.7 The Shearlet Transforms
3.7.1 Shearlet Transform: A Group Theoretical Approach
3.7.2 Shearlet Transform: An Analytical Perspective
3.7.3 Visualization of the Shearlet Coefficients
3.7.4 Cone-adapted Shearlet Transform
3.8 The Bendlet Transform
3.8.1 Definition and Basic Properties
3.8.2 Applications of Bendlet Transform to Partial Differential Equations
3.9 Exercises
4 The Intertwining of Wavelet Transforms
4.1 Introduction
4.2 The Fractional Wavelet Transform
4.2.1 Definition and Basic Properties
4.2.2 Constant Q-Property and Time-Fractional-Frequency Resolution
4.3 The Fractional Stockwell Transform
4.3.1 Definition and Basic Properties
4.4 The Linear Canonical Wavelet Transform
4.4.1 Convolution-based Linear Canonical Wavelet Transform 4.4.2 Constant Q-Property and Resolution of the Convolution-based Linear Canonical Wavelet Transform
4.4.3 Composition of Linear Canonical Wavelet Transforms
4.4.4 Modified Linear Canonical Wavelet Transform
4.5 The Linear Canonical Stockwell Transform
4.5.1 Definition and Basic Properties
4.6 The Linear Canonical Ridgelet Transform
4.6.1 Definition and Basic Properties
4.7 The Linear Canonical Curvelet and Ripplet Transforms
4.7.1 Linear Canonical Curvelet Transform
4.7.2 Linear Canonical Ripplet Transform
4.8 The Linear Canonical Shearlet Transform
4.8.1 Definition and Basic Properties
4.8.2 Uncertainty Principle for the Linear Canonical Shearlet Transform
4.9 Exercises
5 The Wavelet Transforms and Kith 413
5.1 Introduction
5.2 The Laguerre Wavelet Transform
5.2.1 Laguerre Polynomials and Transform
5.2.2 Laguerre Wavelet Transform: Definition and Basic Properties
5.3 The Legendre Wavelet Transform
5.3.1 Legendre Polynomials and Transform
5.3.2 Legendre Wavelet Transform: Definition and Basic Properties
5.4 The Bessel Wavelet Transform
5.4.1 Bessel Functions and Hankel Transform
5.4.2 Bessel Wavelet Transform: Definition and Basic Properties
5.5 The Dunkl Wavelet Transform
5.5.1 Dunkl Transform
5.5.2 Dunkl Wavelet Transform: Definition and Basic Properties
5.6 The Mehler-Fock Wavelet Transform
5.6.1 Mehler-Fock Transform
5.6.2 Mehler-Fock Wavelet Transform: Definition and Basic Properties
5.7 The Interface of Wavelet and Hartley Transforms
5.7.1 Hartley Transform
5.7.2 Hartley-based Wavelet Transform
5.8 Exercises
Bibliography 457
Index 475
Biography
Firdous A. Shah earned his post-graduate degree in pure mathematics from the University of Kashmir and a PhD in applied mathematics from the Central University, Jamia Millia Islamia, New Delhi, India. He is an associate professor of mathematics at the University of Kashmir, South Campus, India. His research interests include the theory of wavelets, time-frequency analysis, abstract harmonic analysis, sampling theory and applications of wavelets to signal processing and mathematical biology. He is the author/co-author of over 150 journal articles and has co-authored two books with Prof. Lokenath Debnath (University of Texas) on wavelet transforms and their applications, published by Birkhauser, Springer. He has received numerous research grants from NBHM and SERB-DST, Govt. of India. He serves as an editorial board member of several reputed journals and is a lifetime member of several academic societies.
Azhar Y. Tantary is a native of Jammu and Kashmir, India. He completed his BSc and MSc degrees in mathematics from the University of Kashmir during the academic sessions 2010-2012 and 2013-2014, respectively. Presently, he is a researcher in the Department of Mathematics, South Campus, University of Kashmir. His broad field of research is wavelet theory and his research interests include integral transforms, sampling theory and mathematical signal processing. He has authored/co-authored more than 15 research articles in different journals of international repute. He has also presented his research at various national and international forums and has received several awards for excellence in research.
