Wavelets Theory and Its Applications

Building on the initial work of researchers such as Haar, Gabor, Ahmed, Zweig, Goupillaud, Morlet, and Grossman, wavelets exploded onto the scientific landscape in the 1980’s and 1990’s, fueled by the growth of computing power.  Wavelet functions and transforms have their analogs in the world of Fourier, with sines and cosines replaced by more exotic wavelets that still have up-down oscillatory behavior, but are often restricted to non-zero values on a finite interval.  Wavelet functions may or may not be smooth.  The visibility of wavelets was accelerated by early applications in areas like audio and image processing and compression, and since Ingrid Daubechies Ten Lectures on Wavelets was published in 1992, a steady series of textbooks on wavelets have been published.

 

Books on wavelets generally fall into one of three categories:  for researchers and Ph.D. candidates; for motivated senior undergraduates, first-year graduate students, and non-mathematician scientists; and for mathematics and science undergraduates with a strong background in linear algebra and coding.  Texts in the first category assume fluency with the Fourier transform, which is used to prove a variety of theorems.  The Fourier transform is used to develop the continuous wavelet transform, while the discrete wavelet transform comes later.  Daubechies’ book, as well as texts by Jansen & Oonincx, and Rao & Bopardikar, fall into this category.  

 

In the second category are books for which proofs are less frequent, so the Fourier transform does not play a large role, at least initially.  These books may include a variety of graphs and diagrams to develop understanding, or specific applications of wavelets in applied mathematics.  The latter half of such books move into more sophisticated mathematical treatments and will feature special topics of interest.  Some books in this category were authored by Strang & Nguyen, Kaiser, Resnikoff & Wells, and Blatter.

 

Books in the third category focus at first on the discrete wavelet transform, especially the transform based on the Haar wavelets, since many of the ideas in wavelet theory can be then introduced using vectors and matrices.  Topics, such as the cascade algorithm to generate wavelet functions, are presented in order to be implemented, and a proof, such that the algorithm converges, may or may not be included.  Van Fleet, Goodman and Jerri have all written books of this type, and there is an older text co-authored by this reviewer.

 

Wavelet Theory and Its Applications – A First Course, by Mani Mehra falls into the second category.  As Mehra writes in the preface, “I want to prepare the students for more advanced books and research articles in the wavelet theory”.  A relatively brief 182 pages, this book breezes through the foundational ideas and formulas of wavelet theory before moving on to advanced topics.  There are few formal proofs, but a number of derivations of formulas are included.  Explanations in the book are brief but clear.

 

The book is in four parts.  Part I (Mathematical Foundations) states some results from linear algebra and functional analysis – types of spaces, inner products, projections, and series of functions, followed by a speedy tour of Fourier analysis.  These foundational ideas are used throughout the book in the explanations, derivations, and proofs that follow.

 

Part II (Introduction to Wavelets) contains a sufficient number of topics for a semester course.  Mehra begins with “Wavelets on Flat Geometries”, with a focus on the real line.  Key ideas are quickly introduced, such as multiresolution analysis, scaling functions, associated dilation equations, and high- and low-pass filter coefficients. Several graphs of wavelet functions are included.  The Fourier transform is applied to generate some formulas, which then leads to multiple methods to generate a wavelet from its filter coefficients or the scaling function.  All of these developments are presented over just ten pages.

 

Continuing through this part of the book, the next topic is the Daubechies wavelets, followed by several other wavelet families (e.g. Coiflet, Shannon, Biorthogonal, Battle-Lemarie).  Next is “Wavelets on Arbitrary Manifolds”, which includes a short discussion of the lifting scheme, followed by an introduction to the continuous and discrete wavelet transforms.  Parts I and II each conclude with a page of exercises.

Parts I and II are a basis for the second half of the book, where Mehra focuses on some applications of wavelets.  In Part III, we read about a number of numerical methods for differential equations, including Wavelet-Galerkin and optimized methods.  Applications of wavelets to inverse problems, turbulence, multigrid methods, and integral equations are all discussed briefly in Part IV of the book.

 

Because wavelets rose the prominence in conjunction with the growth of computing in the latter part of the 20th century, there are good reasons to make connections to computer code in books like these.  Mehra’s book includes some snippets of MATLAB code, linked to the wavelet toolbox in MATLAB, that can be used to implement some of the ideas in the book.  Every chapter also has a wide variety of references, and in several places, Mehra invites the reader to find a reference “for details”.

 

To summarize, the first half of Mehra’s book provides a concise overview of foundational wavelet theory, while the second half focuses on more advanced topics.  The succinctness of the book will appeal to some readers.  For more details, derivations, and context, others will want to supplement this book with online or in-person lectures, or additional readings.

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Edward Aboufadel (aboufade@gvsu.edu) is a Professor of Mathematics, as well as an Associate Vice President for Academic Affairs, at Grand Valley State University in western Michigan.