A Wavelet Tour of Signal Processing: The Sparse Way
Mallat's book is the undisputed reference in this field - it is the only one that covers the essential material in such breadth and depth. - Laurent Demanet, Stanford University
The new edition of this classic book gives all the major concepts, techniques and applications of sparse representation, reflecting the key role the subject plays in today's signal processing. The book clearly presents the standard representations with Fourier, wavelet and time-frequency transforms, and the construction of orthogonal bases with fast algorithms. The central concept of sparsity is explained and applied to signal compression, noise reduction, and inverse problems, while coverage is given to sparse representations in redundant dictionaries, super-resolution and compressive sensing applications.
Features:
* Balances presentation of the mathematics with applications to signal processing
* Algorithms and numerical examples are implemented in WaveLab, a MATLAB toolbox
* Companion website for instructors and selected solutions and code available for students
New in this edition
* Sparse signal representations in dictionaries
* Compressive sensing, super-resolution and source separation
* Geometric image processing with curvelets and bandlets
* Wavelets for computer graphics with lifting on surfaces
* Time-frequency audio processing and denoising
* Image compression with JPEG-2000
* New and updated exercises
A Wavelet Tour of Signal Processing: The Sparse Way, third edition, is an invaluable resource for researchers and R&D engineers wishing to apply the theory in fields such as image processing, video processing and compression, bio-sensing, medical imaging, machine vision and communications engineering.
Stephane Mallat is Professor in Applied Mathematics at École Polytechnique, Paris, France. From 1986 to 1996 he was a Professor at the Courant Institute of Mathematical Sciences at New York University, and between 2001 and 2007, he co-founded and became CEO of an image processing semiconductor company.
Companion website: A Numerical Tour of Signal Processing
application to JPEG 2000 and MPEG-4
Why Read This Book
You should read this book if you want a single, authoritative resource that ties wavelet theory to practical DSP tasks like compression, denoising, and inverse problems. It shows both the mathematical foundations (multiresolution analysis, frames, filter banks) and modern sparse-representation methods that are central to contemporary signal processing.
Who Will Benefit
Graduate students, researchers, and practicing DSP engineers who need a deep, rigorous treatment of wavelets, sparse representations, and their applications to compression, denoising, and inverse problems.
Level: Advanced — Prerequisites: Undergraduate-level linear algebra and calculus, familiarity with Fourier transforms and basic signal processing; basic probability and numerical methods are helpful.
Key Takeaways
- Understand the mathematical foundations of multiresolution analysis and wavelet bases.
- Derive and construct orthogonal and biorthogonal wavelets and their associated filter banks.
- Apply wavelet and time-frequency methods for compression, denoising, and signal representation.
- Use sparse representation concepts (thresholding, matching pursuit, basis pursuit) and redundant dictionaries for efficient modeling.
- Analyze inverse problems, super-resolution and compressive sensing within the wavelet/sparse framework.
- Implement fast discrete wavelet transform algorithms and reason about stability and approximation rates.
Topics Covered
- Introduction and historical context
- Mathematical preliminaries: Fourier analysis and distributions
- Multiresolution analysis and scaling functions
- Orthogonal and biorthogonal wavelet constructions
- Discrete wavelet transform and fast filter-bank algorithms
- Time-frequency representations and frames
- Continuous wavelet transform and applications
- Sparse approximations and nonlinear thresholding
- Redundant dictionaries, matching pursuit, and bases pursuit
- Compression and denoising with wavelets
- Inverse problems, super-resolution, and regularization
- Compressive sensing and sparse recovery
- Numerical issues, implementations, and examples
- Appendices: proofs and mathematical tools
Languages, Platforms & Tools
How It Compares
More comprehensive and modern than Daubechies' 'Ten Lectures on Wavelets' (which is shorter and more introductory); it also complements practical texts focused on implementation by emphasizing mathematical foundations and sparsity theory.
