Sparse and Redundant Representations From Theory to Applications in Signal and Image Processing

ISBN-10: 144197010X

ISBN-13: 9781441970107

Edition: 2010

Authors: Michael Elad
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Book details

Copyright year: 2010
Publisher: Springer
Publication date: 8/19/2010
Binding: Hardcover
Pages: 376
Size: 6.25" wide x 9.25" long x 1.00" tall
Weight: 1.430
Language: English

Michael Elad has been working at The Technion in Haifa, Israel, since 2003 and is currently an Associate Professor. He is one of the leaders in the field of sparse representations. He does prolific research in mathematical signal processing with more than 60 publications in top ranked journals. He is very well recognized and respected in the scientific community.

Sparse and Redundant Representations - Theoretical and Numerical Foundations
Prologue
Underdetermined Linear Systems
Regularization
The Temptation of Convexity
A Closer Look at l<sub>1</sub> Minimization
Conversion of (P<sub>1</sub>) to Linear Programming
Promoting Sparse Solutions
The l<sub>0</sub>-Norm and Implications
The (P<sub>0</sub>) Problem - Our Main Interest
The Signal Processing Perspective
Further Reading
Uniqueness and Uncertainty
Treating the Two-Ortho Case
An Uncertainty Principle
Uncertainty of Redundant Solutions
From Uncertainty to Uniqueness
Uniqueness Analysis for the General Case
Uniqueness via the Spark
Uniqueness via the Mutual-Coherence
Uniqueness via the Babel Function
Upper-Bounding the Spark
Constructing Grassmannian Matrices
Summary
Further Reading
Pursuit Algorithms - Practice
Greedy Algorithms
The Core Idea
The Orthogonal-Matching-Pursuit
Other Greedy Methods
Normalization
Rate of Decay of the Residual in Greedy Methods
Thresholding Algorithm
Numerical Demonstration of Greedy Algorithms
Convex Relaxation Techniques
Relaxation of the l<sub>0</sub>-Norm
Numerical Algorithms for Solving (P<sub>1</sub>)
Numerical Demonstration of Relaxation Methods
Summary
Further Reading
Pursuit Algorithms - Guarantees
Back to the Two-Ortho Case
OMP Performance Guarantee
BP Performance Guarantee
The General Case
OMP Performance Guarantee
Thresholding Performance Guarantee
BP Performance Guarantee
Performance of Pursuit Algorithms - Summary
The Role of the Sign-Pattern
Tropp's Exact Recovery Condition
Summary
Further Reading
From Exact to Approximate Solutions
General Motivation
Stability of the Sparsest Solution
Uniqueness versus Stability - Gaining Intuition
Theoretical Study of the Stability of (P<sub>0</sub><sup>�</sup>)
The RIP and Its Use for Stability Analysis
Pursuit Algorithms
OMP and BP Extensions
Iteratively-Reweighed-Least-Squares (IRLS)
The LARS Algorithm
Quality of Approximations Obtained
The Unitary Case
Performance of Pursuit Algorithms
BPDN Stability Guarantee
Thresholding Stability Guarantee
Summary
Further Reading
Iterative-Shrinkage Algorithms
Background
The Unitary Case - A Source of Inspiration
Shrinkage For the Unitary case
The BCR Algorithm and Variations
Developing Iterative-Shrinkage Algorithms
Surrogate Functions and the Prox Method
EM and Bound-Optimization Approaches
An IRLS-Based Shrinkage Algorithm
The Parallel-Coordinate-Descent (PCD) Algorithm
StOMP: A Variation on Greedy Methods
Bottom Line - Iterative-Shrinkage Algorithms
Acceleration Using Line-Search and SESOP
Iterative-Shrinkage Algorithms: Tests
Summary
Further Reading
Towards Average Performance Analysis
Empirical Evidence Revisited
A Glimpse into Probabilistic Analysis
The Analysis Goals
Two-Ortho Analysis by Candes & Romberg
Probabilistic Uniqueness
Donoho's Analysis
Summary
Average Performance of Thresholding
Preliminaries
The Analysis
Discussion
Summary
Further Reading
The Dantzig-Selector Algorithm
Dantzig-Selector versus Basis-Pursuit
The Unitary Case
Revisiting the Restricted Isometry Machinery
Dantzig-Selector Performance Guaranty
Dantzig-Selector in Practice
Summary
Further Reading
From Theory to Practice - Signal and Image Processing Applications
Sparsity-Seeking Methods in Signal Processing
Priors and Transforms for Signals
The Sparse-Land Model
Geometric Interpretation of Sparse-Land
Processing of Sparsely-Generated Signals
Analysis Versus Synthesis Signal Modeling
Summary
Further Reading
Image Deblurring - A Case Study
Problem Formulation
The Dictionary
Numerical Considerations
Experiment Details and Results
Summary
Further Reading
MAP versus MMSE Estimation
A Stochastic Model and Estimation Goals
Background on MAP and MMSE
The Oracle Estimation
Developing the Oracle Estimator
The Oracle Error
The MAP Estimation
Developing the MAP Estimator
Approximating the MAP Estimator
The MMSE Estimation
Developing the MMSE Estimator
Approximating the MMSE Estimator
MMSE and MAP Errors
More Experimental Results
Summary
Further Reading
The Quest for a Dictionary
Choosing versus Learning
Dictionary-Learning Algorithms
Core Questions in Dictionary-Learning
The MOD Algorithm
The K-SVD Algorithm
Training Structured Dictionaries
The Double-Sparsity Model
Union of Unitary Bases
The Signature Dictionary
Summary
Further Reading
Image Compression - Facial Images
Compression of Facial Images
Previous Work
Sparse-Representation-Based Coding Scheme
The General Scheme
VQ Versus Sparse Representations
More Details and Results
K-SVD Dictionaries
Reconstructed Images
Run-Time and Memory Usage
Comparing to Other Techniques
Dictionary Redundancy
Post-Processing for Deblocking
The Blockiness Artifacts
Possible Approaches For Deblocking
Learning-Based Deblocking Approach
Deblocking Results
Summary
Further Reading
Image Denoising
General Introduction - Image Denoising
The Beginning: Global Modeling
The Core Image-Denoising Algorithm
Various Improvements
From Global to Local Modeling
The General Methodology
Learning the Shrinkage Curves
Learned Dictionary and Globalizing the Prior
The Non-Local-Means Algorithm
3D-DCT Shrinkage: BM3D Denoising
SURE for Automatic Parameter Setting
Development of the SURE
Demonstrating SURE to Global-Threhsolding
Summary
Further Reading
Other Applications
General
Image Separation via MCA
Image = Cartoon + Texture
Global MCA for Image Separation
Local MCA for Image Separation
Image Inpainting and Impulsive Noise Removal
Inpainting Sparse-Land Signals - Core Principles
Inpainting Images - Local K-SVD
Inpainting Images - The Global MCA
Impulse-Noise Filtering
Image Scale-Up
Modeling the Problem
The Super-Resolution Algorithm
Scaling-Up Results
Image Scale-Up: Summary
Summary
Further Reading
Epilogue
What is it All About?
What is Still Missing?
Bottom Line
Notation
Acronyms
Index
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