Nievergelt Y. Wavelets Made Easy

Birkhäuser, 2013. — 303 p.This book explains the nature and computation of mathematical wavelets, which provide a framework and methods for the analysis and the synthesis of signals, images, and other arrays of data. The material presented here addresses the audience of engineers, financiers, scientists, and students looking for explanations of wavelets at the undergraduate level. It requires only a working knowledge or memories of a first course in linear algebra and calculus. The first part of the book answers the following two questions:
What are wavelets? Wavelets extend Fourier analysis.
How are wavelets computed? Fast transforms compute them. To show the practical significance of wavelets, the book also provides transitions into several applications: analysis (detection of crashes, edges, or other events), compression (reduction of storage), smoothing (attenuation of noise), and synthesis (reconstruction after compression or other modification). Such applications include one-dimensional signals (sounds or other time-series), two-dimensional arrays (pictures or maps), and three-dimensional data (spatial diffusion). The applications demonstrated here do not constitute recipes for real implementations, but aim only at clarifying and strengthening the understanding of the mathematics of wavelets.
The second part of the book explains orthogonal projections, discrete and fast Fourier transforms, and Fourier series, as a preparation for the third part and as an answer to the following question:
How are wavelets related to other methods of signal analysis?
The third part of the book invokes occasional results from advanced calculus and focuses on the fol1owing question, which provides a transition into the theory and research on the subject:
How are wavelets designed? (Designs use Fourier transforms.)
More details appear in the chapter summaries on the following page. The material has been taught in various forms for a decade in an undergraduate course at Eastern Washington University, to engineers and students majoring in mathematics or computer science. I thank them for their patience in reading through several drafts.Part A Algorithms for Wavelet Transforms
Haar's Simple Wavelets
Multidimensional Wavelets and Applications
Algorithms for Daubechies Wavelets
Part B Basic Fourier Analysis
Inner Products and Orthogonal Projections
Discrete and Fast Fourier Transforms
Fourier Series for Periodic Functions
Part C Computation and Design of Wavelets
Fourier Transforms on the Line and in Space
Daubechies Wavelets Design
Signal Representations with Wavelets
Part D Directories