Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing
A long long time ago, echoing philosophical and aesthetic principles that existed since antiquity, William of Ockham enounced the principle of parsimony, better known today as Ockham’s razor: “Entities should not be multiplied without neces sity. ” This principle enabled scientists to select the ”best” physical laws and theories to explain the workings of the Universe and continued to guide scienti?c research, leadingtobeautifulresultsliketheminimaldescriptionlength approachtostatistical inference and the related Kolmogorov complexity approach to pattern recognition. However, notions of complexity and description length are subjective concepts anddependonthelanguage“spoken”whenpresentingideasandresults. The?eldof sparse representations, that recently underwent a Big Bang like expansion, explic itly deals with the Yin Yang interplay between the parsimony of descriptions and the “language” or “dictionary” used in them, and it became an extremely exciting area of investigation. It already yielded a rich crop of mathematically pleasing, deep and beautiful results that quickly translated into a wealth of practical engineering applications. You are holding in your hands the ?rst guide book to Sparseland, and I am sure you’ll ?nd in it both familiar and new landscapes to see and admire, as well as ex cellent pointers that will help you ?nd further valuable treasures. Enjoy the journey to Sparseland! Haifa, Israel, December 2009 Alfred M. Bruckstein vii Preface This book was originally written to serve as the material for an advanced one semester (fourteen 2 hour lectures) graduate course for engineering students at the Technion, Israel.
Why Read This Book
You should read this book if you want a unified, practical yet rigorous introduction to sparse representations and dictionary learning as tools for signal and image processing. It teaches both the underlying theory and the concrete algorithms (e.g., OMP, Basis Pursuit, K-SVD) and shows how to apply them to denoising, inpainting and other inverse problems.
Who Will Benefit
Practicing signal-processing engineers, graduate students, and researchers who need to apply sparse coding and dictionary-learning methods to imaging, audio, or inverse problems.
Level: Advanced — Prerequisites: Linear algebra (vector spaces, norms, SVD), basic optimization (convexity, L1 minimization), probability/statistics, and familiarity with standard DSP concepts; MATLAB experience helps.
Key Takeaways
- Understand the mathematical foundations of sparse representations and why sparsity is a useful prior in signal processing.
- Apply pursuit algorithms (Matching Pursuit, OMP, Basis Pursuit) and know their trade-offs in practice.
- Implement and use dictionary learning methods (notably K-SVD) to build adaptive, overcomplete dictionaries.
- Formulate and solve sparse-regularized inverse problems for denoising, deblurring, inpainting, and super-resolution.
- Relate sparse representations to compressed sensing and recognize conditions for uniqueness and stable recovery.
- Evaluate algorithmic performance and numerical considerations when deploying sparse methods on real data.
Topics Covered
- Introduction and Motivation for Sparse Representations
- Sparse Models and Signal Representations
- Measures of Sparsity, Uniqueness and Stability
- Greedy Pursuit Algorithms (MP, OMP) and Variants
- Convex Relaxation and L1-Based Methods (Basis Pursuit)
- Theoretical Guarantees: Coherence, RIP, and Recovery Conditions
- Overcomplete Dictionaries and Transform Models
- Dictionary Learning: K-SVD and Other Algorithms
- Applications I: Image Denoising and Restoration
- Applications II: Inpainting, Super-resolution, and Classification
- Connections to Compressed Sensing and Multi-scale Representations
- Numerical Issues, Implementation Details, and Case Studies
- Appendices: Optimization Tools and MATLAB Examples
Languages, Platforms & Tools
How It Compares
More application- and algorithm-focused than the mathematically rigorous compressive-sensing texts (e.g., Foucart & Rauhut), and more practical for engineers than Eldar & Kutyniok's edited volumes which emphasize theory and broad surveys.
