Wavelets: A Concise Guide
Introduced nearly three decades ago as a variable resolution alternative to the Fourier transform, a wavelet is a short oscillatory waveform for analysis of transients. The discrete wavelet transform has remarkable multi-resolution and energy-compaction properties. Amir-Homayoon Najmi’s introduction to wavelet theory explains this mathematical concept clearly and succinctly.
Wavelets are used in processing digital signals and imagery from myriad sources. They form the backbone of the JPEG2000 compression standard, and the Federal Bureau of Investigation uses biorthogonal wavelets to compress and store its vast database of fingerprints. Najmi provides the mathematics that demonstrate how wavelets work, describes how to construct them, and discusses their importance as a tool to investigate and process signals and imagery. He reviews key concepts such as frames, localizing transforms, orthogonal and biorthogonal bases, and multi-resolution. His examples include the Haar, the Shannon, and the Daubechies families of orthogonal and biorthogonal wavelets.
Our capacity and need for collecting and transmitting digital data is increasing at an astonishing rate. So too is the importance of wavelets to anyone working with and analyzing digital data. Najmi’s primer will be an indispensable resource for those in computer science, the physical sciences, applied mathematics, and engineering who wish to obtain an in-depth understanding and working knowledge of this fascinating and evolving field.
Why Read This Book
You should read this book if you want a compact, engineer-friendly introduction to wavelet theory and practice: it explains the core mathematics, shows how discrete wavelet transforms are implemented with filter banks, and connects the theory to real applications such as JPEG2000 and denoising.
Who Will Benefit
Graduate students, practicing DSP engineers, and developers who need a focused, practical primer on wavelets for signal and image processing tasks.
Level: Intermediate — Prerequisites: Basic linear algebra, familiarity with Fourier methods and discrete-time signals, and some exposure to digital filtering and convolution.
Key Takeaways
- Understand the theoretical foundations of continuous and discrete wavelet transforms and multiresolution analysis.
- Construct and classify common wavelet families (Haar, Daubechies, Symlets, Coiflets) and biorthogonal wavelets.
- Implement fast discrete wavelet transforms using two-channel filter-bank structures and convolution/downsampling.
- Apply wavelet techniques to practical problems such as image compression (JPEG2000) and denoising via thresholding.
- Analyze time-frequency localization and energy compaction properties of wavelets for transient signal analysis.
- Use wavelet packets and basis selection strategies for more flexible signal representations.
Topics Covered
- 1. Introduction and Historical Motivation
- 2. Time-Frequency Localization and Short-Time Analysis
- 3. The Continuous Wavelet Transform
- 4. Multiresolution Analysis (MRA) and Scaling Functions
- 5. Discrete Wavelet Transform (DWT) and the Fast Wavelet Transform
- 6. Two-Channel Filter Banks and Perfect Reconstruction
- 7. Construction of Orthogonal and Biorthogonal Wavelets
- 8. Prominent Wavelet Families (Haar, Daubechies, Symlets, Coiflets)
- 9. Wavelet Packets and Best-Basis Selection
- 10. Two-Dimensional Wavelets and Image Processing
- 11. Compression (JPEG2000) and Practical Implementations
- 12. Denoising, Thresholding, and Other Signal-Processing Applications
- 13. Appendices: Useful Mathematical Background and Tables
Languages, Platforms & Tools
How It Compares
More concise and application-focused than Stephane Mallat's A Wavelet Tour of Signal Processing, and far less mathematically deep than Ingrid Daubechies' Ten Lectures on Wavelets; useful as a compact engineer's primer.
