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    Cover
    Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing
    Copyright
    9781441970107
    Foreword
    Preface
    Acknowledgments
    Contents
    Part I Sparse and Redundant Representations – Theoretical and Numerical Foundations
    Chapter 1 Prologue
    1.1 Underdetermined Linear Systems
    1.2 Regularization
    1.3 The Temptation of Convexity
    1.4 A Closer Look at l1 Minimization
    1.5 Conversion of (P1) to Linear Programming
    1.6 Promoting Sparse Solutions
    1.7 The l0-Norm and Implications
    1.8 The (P0) Problem – Our Main Interest
    1.9 The Signal Processing Perspective
    Further Reading
    Chapter 2 Uniqueness and Uncertainty
    2.1 Treating the Two-Ortho Case
    2.1.1 An Uncertainty Principle
    2.1.2 Uncertainty of Redundant Solutions
    2.1.3 From Uncertainty to Uniqueness
    2.2 Uniqueness Analysis for the General Case
    2.2.1 Uniqueness via the Spark
    2.2.2 Uniqueness via the Mutual-Coherence
    2.2.3 Uniqueness via the Babel Function
    2.2.4 Upper-Bounding the Spark
    2.3 Constructing Grassmannian Matrices
    2.4 Summary
    Further Reading
    Chapter 3 Pursuit Algorithms – Practice
    3.1 Greedy Algorithms
    3.1.1 The Core Idea
    3.1.2 The Orthogonal-Matching-Pursuit
    3.1.3 Other Greedy Methods
    3.1.4 Normalization
    3.1.5 Rate of Decay of the Residual in Greedy Methods
    3.1.6 Thresholding Algorithm
    3.1.7 Numerical Demonstration of Greedy Algorithms
    3.2 Convex Relaxation Techniques
    3.2.1 Relaxation of the l0-Norm
    3.2.2 Numerical Algorithms for Solving (P1)
    3.2.3 Numerical Demonstration of Relaxation Methods
    3.3 Summary
    Further Reading
    Chapter 4 Pursuit Algorithms – Guarantees
    4.1 Back to the Two-Ortho Case
    4.1.1 OMP Performance Guarantee
    4.1.2 BP Performance Guarantee
    4.2 The General Case
    4.2.1 OMP Performance Guarantee
    4.2.2 Thresholding Performance Guarantee
    4.2.3 BP Performance Guarantee
    4.2.4 Performance of Pursuit Algorithms – Summary
    4.3 The Role of the Sign-Pattern
    4.4 Tropp’s Exact Recovery Condition
    4.5 Summary
    Further Reading
    Chapter 5 From Exact to Approximate Solutions
    5.1 General Motivation
    5.2 Stability of the Sparsest Solution
    5.2.1 Uniqueness versus Stability – Gaining Intuition
    5.2.2 Theoretical Study of the Stability of (P0)
    5.2.3 The RIP and Its Use for Stability Analysis
    5.3 Pursuit Algorithms
    5.3.1 OMP and BP Extensions
    5.3.2 Iteratively-Reweighed-Least-Squares (IRLS)
    5.3.3 The LARS Algorithm
    5.3.4 Quality of Approximations Obtained
    5.4 The Unitary Case
    5.5 Performance of Pursuit Algorithms
    5.5.1 BPDN Stability Guarantee
    5.5.2 Thresholding Stability Guarantee
    5.6 Summary
    Further Reading
    Chapter 6 Iterative-Shrinkage Algorithms
    6.1 Background
    6.2 The Unitary Case - A Source of Inspiration
    6.2.1 Shrinkage For the Unitary case
    6.2.2 The BCR Algorithm and Variations
    6.3 Developing Iterative-Shrinkage Algorithms
    6.3.1 Surrogate Functions and the Prox Method
    6.3.2 EM and Bound-Optimization Approaches
    6.3.3 An IRLS-Based Shrinkage Algorithm
    6.3.4 The Parallel-Coordinate-Descent (PCD) Algorithm
    6.3.5 StOMP: A Variation on Greedy Methods
    6.3.6 Bottom Line – Iterative-Shrinkage Algorithms
    6.4 Acceleration Using Line-Search and SESOP
    6.5 Iterative-Shrinkage Algorithms: Tests
    6.6 Summary
    Further Reading
    Chapter 7 Towards Average Performance Analysis
    7.1 Empirical Evidence Revisited
    7.2 A Glimpse into Probabilistic Analysis
    7.2.1 The Analysis Goals
    7.2.2 Two-Ortho Analysis by Candes & Romberg
    7.2.3 Probabilistic Uniqueness
    7.2.4 Donoho’s Analysis
    7.2.5 Summary
    7.3 Average Performance of Thresholding
    7.3.1 Preliminaries
    7.3.2 The Analysis
    7.3.3 Discussion
    7.4 Summary
    Further Reading
    Chapter 8 The Dantzig-Selector Algorithm
    8.1 Dantzig-Selector versus Basis-Pursuit
    8.2 The Unitary Case
    8.3 Revisiting the Restricted Isometry Machinery
    8.4 Dantzig-Selector Performance Guaranty
    8.5 Dantzig-Selector in Practice
    8.6 Summary
    Further Reading
    Part II From Theory to Practice – Signal and Image Processing Applications
    Chapter 9 Sparsity-Seeking Methods in Signal Processing
    9.1 Priors and Transforms for Signals
    9.2 The Sparse-Land Model
    9.3 Geometric Interpretation of Sparse-Land
    9.4 Processing of Sparsely-Generated Signals
    9.5 Analysis Versus Synthesis Signal Modeling
    9.6 Summary
    Further Reading
    Chapter 10 Image Deblurring – A Case Study
    10.1 Problem Formulation
    10.2 The Dictionary
    10.3 Numerical Considerations
    10.4 Experiment Details and Results
    10.5 Summary
    Further Reading
    Chapter 11 MAP versus MMSE Estimation
    11.1 A Stochastic Model and Estimation Goals
    11.2 Background on MAP and MMSE
    11.3 The Oracle Estimation
    11.3.1 Developing the Oracle Estimator
    11.3.2 The Oracle Error
    11.4 The MAP Estimation
    11.4.1 Developing the MAP Estimator
    11.4.2 Approximating the MAP Estimator
    11.5 The MMSE Estimation
    11.5.1 Developing the MMSE Estimator
    11.5.2 Approximating the MMSE Estimator
    11.6 MMSE and MAP Errors
    11.7 More Experimental Results
    11.8 Summary
    Further Reading
    Chapter 12 The Quest for a Dictionary
    12.1 Choosing versus Learning
    12.2 Dictionary-Learning Algorithms
    12.2.1 Core Questions in Dictionary-Learning
    12.2.2 The MOD Algorithm
    12.2.3 The K-SVD Algorithm
    12.3 Training Structured Dictionaries
    12.3.1 The Double-Sparsity Model
    12.3.2 Union of Unitary Bases
    12.3.3 The Signature Dictionary
    12.4 Summary
    Further Reading
    Chapter 13 Image Compression – Facial Images
    13.1 Compression of Facial Images
    13.2 Previous Work
    13.3 Sparse-Representation-Based Coding Scheme
    13.3.1 The General Scheme
    13.3.2 VQ Versus Sparse Representations
    13.4 More Details and Results
    13.4.1 K-SVD Dictionaries
    13.4.2 Reconstructed Images
    13.4.3 Run-Time and Memory Usage
    13.4.4 Comparing to Other Techniques
    13.4.5 Dictionary Redundancy
    13.5 Post-Processing for Deblocking
    13.5.1 The Blockiness Artifacts
    13.5.2 Possible Approaches For Deblocking
    13.5.3 Learning-Based Deblocking Approach
    13.6 Deblocking Results
    13.7 Summary
    Further Reading
    Chapter 14 Image Denoising
    14.1 General Introduction – Image Denoising
    14.2 The Beginning: Global Modeling
    14.2.1 The Core Image-Denoising Algorithm
    14.2.2 Various Improvements
    14.3 From Global to Local Modeling
    14.3.1 The General Methodology
    14.3.2 Learning the Shrinkage Curves
    14.3.3 Learned Dictionary and Globalizing the Prior
    14.3.4 The Non-Local-Means Algorithm
    14.3.5 3D-DCT Shrinkage: BM3D Denoising
    14.4 SURE for Automatic Parameter Setting
    14.4.1 Development of the SURE
    14.4.2 Demonstrating SURE to Global-Threhsolding
    14.5 Summary
    Further Reading
    Chapter 15 Other Applications
    15.1 General
    15.2 Image Separation via MCA
    15.2.1 Image = Cartoon + Texture
    15.2.2 Global MCA for Image Separation
    15.2.3 Local MCA for Image Separation
    15.3 Image Inpainting and Impulsive Noise Removal
    15.3.1 Inpainting Sparse-Land Signals – Core Principles
    15.3.2 Inpainting Images – Local K-SVD
    15.3.3 Inpainting Images – The Global MCA
    15.3.4 Impulse-Noise Filtering
    15.4 Image Scale-Up
    15.4.1 Modeling the Problem
    15.4.2 The Super-Resolution Algorithm
    15.4.2.1 Training Phase
    15.4.2.2 Testing Phase: Scaling-Up an Image
    15.4.3 Scaling-Up Results
    15.4.4 Image Scale-Up: Summary
    15.5 Summary
    Further Reading
    Chapter 16 Epilogue
    16.1 What is it All About?
    16.2 What is Still Missing?
    16.3 Bottom Line
    Notation
    Acronyms
    Index
    Sparse and Redundant Representations
    Sparse and Redundant Representations From Theory to Applications in Signal and Image Processing Michael Elad The Technion – Israel Institute of Technology Haifa, Israel
    Michael Elad The Technion – Israel Institute of Technology Computer Science Department 32000 Haifa, Israel elad@cs.technion.ac.il e-ISBN 978-1-4419-7011-4 ISBN 978-1-4419-7010-7 DOI 10.1007/978-1-4419-7011-4 Springer New York Dordrecht Heidelberg London L ibrary of Congress Control Number: 2010933513 Mathematics Subject Classification (2010): 94A12, 62H35, 62M40, 68U10, 94A08, 94H60, 46N10 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
    To my loving and devoted family, My parents, Rita and Shmuel Elbaz, who have shown me the way, My children Sharon, Adar, and Yuval, who give purpose to this way, and My wife, Ritta, who constantly paves this way with love and understanding.
    Foreword 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 scientific research, leading to beautiful results like the minimal description length approach to statistical inference and the related Kolmogorov complexity approach to pattern recognition. However, notions of complexity and description length are subjective concepts and depend on the language “spoken” when presenting ideas and results. The field of 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 first guide-book to Sparseland, and I am sure you’ll find in it both familiar and new landscapes to see and admire, as well as ex- cellent pointers that will help you find further valuable treasures. Enjoy the journey to Sparseland! Haifa, Israel, December 2009 Alfred M. Bruckstein vii
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