A Primer on Wavelets and Their Scientific Applications (Studies in Advanced Mathematics)
In the first edition of his seminal introduction to wavelets, James S. Walker informed us that the potential applications for wavelets were virtually unlimited. Since that time thousands of published papers have proven him true, while also necessitating the creation of a new edition of his bestselling primer. Updated and fully revised to include the latest developments, this second edition of A Primer on Wavelets and Their Scientific Applications guides readers through the main ideas of wavelet analysis in order to develop a thorough appreciation of wavelet applications.
Ingeniously relying on elementary algebra and just a smidgen of calculus, Professor Walker demonstrates how the underlying ideas behind wavelet analysis can be applied to solve significant problems in audio and image processing, as well in biology and medicine.
Nearly twice as long as the original, this new edition provides
· 104 worked examples and 222 exercises, constituting a veritable book of review material
· Two sections on biorthogonal wavelets
· A mini-course on image compression, including a tutorial on arithmetic compression
· Extensive material on image denoising, featuring a rarely covered technique for removing isolated, randomly positioned clutter
· Concise yet complete coverage of the fundamentals of time-frequency analysis, showcasing its application to audio denoising, and musical theory and synthesis
· An introduction to the multiresolution principle, a new mathematical concept in musical theory
· Expanded suggestions for research projects
· An enhanced list of references
· FAWAV: software designed by the author, which allows readers to duplicate described applications and experiment with other ideas.
To keep the book current, Professor Walker has created a supplementary website. This online repository includes ready-to-download software, and sound and image files, as well as access to many of the most important papers in the field.
Why Read This Book
You will get a clear, intuitive development of wavelet ideas without needing heavy functional analysis, plus concrete techniques (filter banks, fast DWT) you can apply to denoising, compression, and time-frequency analysis. The book balances theory and examples so you can build practical signal-processing solutions and understand why they work.
Who Will Benefit
Graduate students and practicing engineers in DSP, audio/image processing, and scientific computing who want a practical, intuitive introduction to wavelets and how to apply them to real problems.
Level: Intermediate — Prerequisites: Basic calculus and linear algebra; familiarity with Fourier transforms and basic signals-and-systems concepts.
Key Takeaways
- Explain the fundamentals of continuous and discrete wavelet transforms and their time-frequency interpretation.
- Construct multiresolution analyses and derive scaling functions and wavelets (including Haar and Daubechies examples).
- Implement the fast wavelet transform using two-channel filter banks and understand practical sampling/aliasing issues.
- Design or choose orthogonal and biorthogonal wavelet bases for specific tasks.
- Apply wavelets to practical problems such as denoising, compression, and feature extraction in signals and images.
Topics Covered
- Introduction and motivation: shortcomings of Fourier methods
- Mathematical preliminaries: basic transforms and function spaces
- The continuous wavelet transform (CWT) and time-frequency localization
- Multiresolution analysis (MRA) and scaling functions
- Construction of orthonormal wavelet bases (Haar, Daubechies)
- Filter banks and the Mallat algorithm (fast DWT)
- Biorthogonal wavelets and spline wavelets
- Wavelet packets and best-basis ideas
- Practical issues: boundary handling, sampling, and numerical stability
- Applications: denoising, compression, and signal detection
- Applications in image processing and scientific data analysis
- Implementation notes and examples
- Appendices: proofs and additional mathematical background
How It Compares
More accessible and application-oriented than Daubechies' Ten Lectures (which is more mathematical), and more concise and introductory than Mallat's A Wavelet Tour of Signal Processing (which is broader and deeper).
