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Spie Press Book

Signal and Image Restoration: Information-Theoretic Approaches
Author(s): Joseph P. Noonan; Prabahan Basu
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Book Description

The goal of this book is to present a unified information-theoretic approach to solve estimation problems commonly faced in signal processing applications. The authors provide new approaches to this problem, as well as new interpretations for existing techniques. Major applications of this work include image restoration, communication channel estimation, text restoration, and system modeling. A general approach to solving a number of detection and estimation problems utilizing concepts from information theory is developed, and the theoretical development of the approach—as well as important applications—is given.

Book Details

Date Published: 28 December 2011
Pages: 116
ISBN: 9780819488213
Volume: PM213

Table of Contents
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Preface
Part I Preliminaries
1 Information Theory Preliminaries
1.1 Scope
1.2 History
    1.2.1 Background
    1.2.2 Definitions and concepts
2 The Inverse Problem
2.1 Introduction
2.2 Signal Restoration
2.3 Well-Posed and Ill-Posed Problems
2.4 Naïve Approaches to Inverse Problems
2.5 Conclusion
3 Review of Signal Restoration Techniques
3.1 Introduction
3.2 Wiener-Kolmogorov Filters
3.3 Constrained Least-Squares Restoration
3.4 Bayesian Restoration
3.5 Maximum Entropy Reconstruction
3.6 The Iterative Technique
    3.6.1 Van Cittert Iteration
3.7 Conclusion
Part II Density Estimation: Maximum Entropy
4 Maximum-Entropy Density Estimation
4.1 Introduction
4.2 Density Estimation
    4.2.1 Information-driven estimation
4.3 Maximum Entropy Estimation
4.4 Estimating Channel Tap Densities Using Maximum Entropy
4.5 Conclusion
5 Error Bounds for Maximum-Entropy Estimates
5.1 Introduction
5.2 Preliminaries
    5.2.1 Legendre transforms
    5.2.2 Exponential families
    5.2.3 Dual representations of exponential family members
    5.2.4 Divergence distance
    5.2.5 Information projection
5.3 Properties of Entropy Estimates
    5.3.1 Maximum entropy and maximum likelihood
    5.3.2 Invariance property
5.4 Estimation Error Bounds
    5.4.1 Overspecified models
    5.4.2 Underspecified models
5.5 Data-Driven Estimation
5.6 Conclusion
Part III Point Estimation: Signal Restoration
6 The Stabilizing-Functional Approach to Regularization
6.1 Introduction
6.2 Regularization Theory
6.3 The Stabilizing Functional Approach
    6.3.1 Mathematical formulation
6.4 Conclusion
7 Applying Information-Theoretic Distances and the GMF
7.1 Introduction
    7.1.1 Definition of the signal space
    7.1.2 The set of feasible solutions
7.2 Mutual Information Signal Restoration
    7.2.1 Mutual information
    7.2.2 Minimizing the stabilizing functional
    7.2.3 Convergence of the GMF
        7.2.3.1 Convergence criteria—strong condition
        7.2.3.2 Problem-specific weaker conditions—illustration
    7.2.4 Incorporating hard constraints
    7.2.5 Making λ Adaptive
7.3 Conclusion
8 Special Cases of the GMF
8.1 Introduction
8.2 Gradient-Based Technique
8.3 Least Squares as a Special Case
8.4 Maximum Likelihood as a Special Case
8.5 Convergence of the Particular Variants
8.6 Conclusion
9 Applications
9.1 Introduction
9.2 Spectral Estimation
    9.2.1 Problem statement and model
    9.2.2 Approach
    9.2.3 Applications and examples
        9.2.3.1 Example 1
        9.2.3.2 Example 2
9.3 Image Restoration
    9.3.1 Model
    9.3.2 Approach
    9.3.3 Results
9.4 Text Image Restoration
    9.4.1 Problem statement and model
    9.4.2 Defocus and low resolution
    9.4.3 Approach
        9.4.3.1 The Radon transform
        9.4.3.2 MIM algorithm
    9.4.4 Super-resolution and deblurring algorithm
    9.4.5 Results
9.5 Conclusion
Appendix A Generalized Projections
A.1 Popular Classes of Distances
    A.1.1 Euclidian distance
    A.1.2 Two families of distances
    A.1.3 Examples
A.2 Generalized Projections
Appendix B Definitions from Linear Algebra
Appendix C Definitions from Analysis
Appendix D Notation
Appendix E Acronyms
Index

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