Introduction to Wavelets (original) (raw)

Wavelets and Wavelet Transform Systems and Their Applications

2022

Dedicated to my family: My Wife Caroline and my Children Obinna and Chijioke. I also dedicate this book to all my students, particularly all of my many master's and doctoral students from all over the world. Preface Overview This book is all about wavelets, wavelet transforms, and how they can be applied to solve problems in different fields of study. The question asked often is, what are wavelets? The answer is that wavelets are waveforms of limited duration that have average values of zero. In comparison to sinusoids, wavelets do have a beginning and an end, while sinusoids theoretically extend from minus to plus infinity. Sinusoids are smooth and predictable and are good at describing constant frequency which otherwise can be called stationary signals. In the case of wavelets, they are irregular, of limited duration, and often non-symmetrical. They are better at describing anomalies, pulses, and other events that start and stop within the signal. This book on wavelets and wavelet transform systems and their applications has grown out of my teaching "Wavelets and Their Applications" graduate course and my research activities in the fields of digital signal processing and communication systems for many decades. The notes on which this book is based on have been used for a one-semester graduate course entitled "Wavelets and Their Applications" that I have taught for several decades at Prairie View A & M University. The book chapters have increased to 21 because of additional new materials considered and therefore can be used for a two-semester course as well. The materials have been updated continuously because of active research in the application of wavelets and wavelet transforms to several areas of science and engineering and lots of research with my graduate students in the areas of wavelet applications. We live in the Information Age where information is analyzed, synthetized, and stored at a much faster rate using different techniques such as different wavelets and wavelet transforms. For many decades, wavelets and wavelet transforms have received much attention in the literature of many communities in the areas of science and engineering. There are different types of wavelets. These wavelets are used as analyzing tools by pure mathematicians (in harmonic analysis, for the study of vii Calderon-Zygmund operators), by statisticians (in nonparametric estimation), and by electrical engineers (in signal analysis). In physics, wavelets are used because of their applications to time-frequency or phase-space analysis and their renormalization concepts. In computer vision research, wavelets are used for "scale-space" methods. In stochastic processes, they are used in application of self-similar processes. Because of wavelets and wavelet transforms' connections with multirate filtering, quadrature mirror filters, and sub-band coding, they have found home in the digital signal processing community. The image processing community uses wavelets because of their applications in pyramidal image representation and compression. In harmonic analysis, wavelets are used because of the special properties of wavelet bases, while the speech processing community uses wavelets because of their efficient signal representation, event extraction, and the mimicking of the human auditory system. Most importantly, wavelet analysis tools can be used as an adaptable exchange to Fourier transform analysis and representation. While there may be many books written over the past decades in the area of wavelets, it is hard to find a wavelet book that is not heavily into rigorous mathematical equations, and in most cases, little or no real applications. In addition, most of the books are not written as textbooks for classroom teaching and to make students understand what wavelets are and how to apply them to solving societal problems. In this book, it is very simplified to real applications in solving societal problems. It is not buried into rigorous mathematical formulas. The book is very suitable as a textbook for upper-level undergraduate study and graduate studies. The practicing engineers in industry will find the book very useful. Not only can the book be used for training of future digital signal processing engineers, it can also be used in research, developing efficient and faster computational algorithms for different multidisciplinary applications. Engineering and scientific professionals can use this book in their research and work-related activities. In actuality, wavelets provide a common link between mathematicians and engineers. Topics such as decomposition and reconstruction algorithms, subdivision algorithms, fast numerical computations, frames, time-frequency localizations, and continuous-and discrete-wavelet transforms are covered for their use of wavelets and wavelet transforms. In addition, topics such as fractals and fractal transforms, mixed signal systems, sub-band coding, image compression, real-time filtering, radar applications, transient analysis, medical imaging, segmentation, blockchain systems, information security, and vibration in aeroelastic systems are some of the areas covered in the book. Applications of many of these wavelets and wavelet transform analyses are developed across disciplines in the book. This book, entitled Wavelets and Wavelet Transform Analysis and Applications: A Signal Processing Approach is a unique book because of its in-depth treatment of the applications of wavelets and wavelet transforms in many areas across many disciplines. The book does this in a very simplified and understandable manner without the mathematical rigor that scares many people away from the field. It uses lots of diagrams to illustrate points being discussed. In addition, the concepts introduced in the book are reinforced with review questions and problems. MATLAB codes and algorithms viii Preface Preface ix x Preface I thank Prairie View A&M University (PVAMU) and The Texas A&M University Board of Regents that approved the Center of Excellence for Communication Systems Technology Research (CECSTR) as one of the Board of Regents Approved Centers on the Campus of PVAMU and within the entire Texas A&M System. I thank all the student researchers and faculty colleagues at CECSTR and the College of Engineering, especially my colleagues in the Department of Electrical and Computer Engineering for all their support. My special thanks go to our former president, Dr. George Wright; former provost, Dr. Thomas-Smith; and former vice president for research and a great friend Dr. Willie Trotty. Special thanks to my former dean of the College of Engineering, Dean Bryant, for all his support. Without the help and support of these giants, there would not be CECSTR, and therefore there would not have been the kind of research work that may have resulted in some of the information in this book. Finally, I thank my wife Caroline Chioma Akujuobi and my two sons Obinna and Chijioke Akujuobi for their patience, encouragement, and support throughout the time I was preparing the manuscript for this book.

The mathematical theory of wavelets

… analysis—a celebration (Il Ciocco, 2000), 2001

We present an overview of some aspects of the mathematical theory of wavelets. These notes are addressed to an audience of mathematicians familiar with only the most basic elements of Fourier Analysis. The material discussed is quite broad and covers several topics involving wavelets. Though most of the larger and more involved proofs are not included, complete references to them are provided. We do, however, present complete proofs for results that are new (in particular, this applies to a recently obtained characterization of "all" wavelets in section 4).

An Introduction to Wavelet Transform

Wavelet transforms have become one of the most important and powerful tool of signal representation. Nowadays, it has been used in image processing, data compression, and signal processing. This paper will introduce the basic concept for Wavelet Transforms, the fast algorithm of Wavelet Transform, and some applications of Wavelet Transform. The difference between conventional Fourier Transform and modern time-frequency analysis will also be discussed.

Wavelets with Application in Image Compression

Emerging Technologies in Intelligent Applications for Image and Video Processing

This chapter focus mainly on different wavelet transform algorithms as Burt's Pyramid, Mallat's Pyramidal Algorithm, Feauveau's non dyadic structure and its application in Image compression. This chapter focus on mathematical concepts involved in wavelet transform like convolution, scaling function, wavelet function, Multiresolution analysis, inner product etc., and how these mathematical concepts are liked to image transform application. This chapter gives an idea towards wavelets and wavelet transforms. Image compression based on wavelet transform consists of transform, quantization and encoding. Basic focus is not only on transform step, selection of particular wavelet, wavelets involved in new standard of image compression but also on quantization and encoding, Huffman code, run length code. Difference in between JPEG and JPEG2000, Quantization and sampling, wavelet function and wavelet transform are also given. This chapter is also giving some basic idea of MATLAB to assist readers in understanding MATLAB Programming in terms of image processing.

Wavelets and its Various Uses in Image Processing

2014

Along with the development of information technology, digital signal have ruled the world of signal processing. This development in digital signal processing has motivated research in analog to digital transformations in various fields. Digital image processing is also one such field. Wavelet transform has become one of the most powerful tool of signal representation. Nowadays, it has been used in image processing, data compression, and signal processing. This paper introduces basic concepts for Wavelet Transforms and some applications of Wavelet Transform.

WAVELETS AND THEIR APPLICATION IN IMAGE PROCESSING WITH FUTURE RESEARCH PROSPECT

The past decade has witnessed the development of wavelet analysis, a new tool that emerged from mathematics and was quickly adopted by diverse fields of science and engineering. In the brief period since its creation in 1987-88, it has reached a certain level of maturity as a well-defined mathematical discipline, with its own conferences, journals, research monographs, and textbooks proliferating at a rapid rate. Wavelet analysis has begun to play a serious role in a broad range of applications, including signal processing, data and image compression, solution of partial differential equations, modelling multiscale phenomena, and statistics. There seem to be no limits to the subjects where it may have utility.Our aim is to explore some additional topics that extend the basic ideas of wavelet analysis KEYWORDS: discrete wavelet transform , image processing , noise removal , median filter.