Preface
Contents
About the Authors
1 Introduction
1.1 Implications for Image Restoration
1.2 Regularization Methods for Image Restoration
1.2.1 Image Degradation Mechanisms and Degradation Modeling
1.2.2 Regularization Methods Based on Variational Partial Differential Equations
1.2.3 Regularization Methods Based on Wavelet Frame Theory
1.2.4 Regularization Methods Based on Sparse Representation of Images
1.2.5 Random Field-Based Regularization Methods
1.3 Nonlinear Iterative Algorithm for Image Restoration
1.3.1 Traditional Methods
1.3.2 Operator Splitting Methods
1.3.3 Convergence Analysis of the Splitting Algorithms
1.3.4 Adaptive Estimation of the Regularization Parameter
References
2 Mathematical Fundamentals
2.1 Summarize
2.2 Convolution
2.2.1 One-Dimensional Discrete Convolution
2.2.2 Two-Dimensional Discrete Convolution
2.3 Fourier Transform and Discrete Fourier Transform
2.4 Theory and Methods of Fixed-Points in Hilbert Spaces
2.4.1 Hilbert Space
2.4.2 Non-expansive Operators with Fixed-Point Iterations
2.4.3 Maximally Monotone Operator
2.4.4 Solution of the l1-ball Projection Problem
Reference
3 Ill-Poseness of Imaging Inverse Problems and Regularization for Detail Preservation
3.1 Summarize
3.2 Typical Types of Image Blur
3.3 The Ill-Posed Nature of Image Deblurring
3.3.1 Discretization of Convolution Equations and Ill-Posed Analysis of Blur Matrices
3.3.2 Image Restoration Based on Inverse Filter
3.4 Tikhonov Image Regularization
3.4.1 Tikhonov Regularization Idea
3.4.2 Wiener Filtering
3.4.3 Constrained Least Square Filtering
3.5 Detail-Preserving Regularization for Image
3.5.1 Total Generalized Variational Regularization Model
3.5.2 Shearlet Regularization Model
3.6 Image Quality Evaluation
References
4 Fast Parameter Estimation in TV-Based Image Restoration
4.1 Summarize
4.2 Overview of Adaptive Parameter Estimation Methods in TV Image Restoration
4.3 Fast Adaptive Parameter Estimation Based on ADMM and Discrepancy Principle
4.3.1 Augmented Lagrangian Model for TV Regularized Problem
4.3.2 Algorithm Derivation
4.3.3 Convergence Analysis
4.3.4 Parameter Settings
4.4 Extension of Fast Adaptive Parameter Estimation Algorithm
4.4.1 Equivalent Splitting Bregman Algorithm
4.4.2 Interval Constrained TV Image Restoration with Fast Adaptive Parameter Estimation
4.5 Experimental Results
4.5.1 Experiment 1: Implications for Significance Regularization Parameter Estimation
4.5.2 Experiment 2—Comparison with Other Adaptive Algorithms
4.5.3 Experiment 3—Comparison of Denoising Experiments
References
5 Parallel Alternating Derection Method of Multipliers with Application to Image Restoration
5.1 Summarize
5.2 Parallel Alternating Direction Method of Multipliers
5.2.1 A General Description of the Regularized Image Restoration Objective Function
5.2.2 Augmented Lagrangian Function with Saddle Point Condition
5.2.3 Algorithm Derivation
5.3 Convergence Analysis
5.3.1 Convergence Proof
5.3.2 Convergence Rate Analysis
5.4 Application of PADMM to TGV/Shearlet Compound Regularized Image Restoration
5.5 Experimental Results
5.5.1 Grayscale Image Deblurring Experiment
5.5.2 RGB Image Deblurring Experiment
5.5.3 MRI Reconstruction Experiment
References
6 Parallel Primal-dual Method with Application to Image Restoration
6.1 Summarize
6.2 Parallel Primal-dual Splitting Method
6.2.1 A General Description of the Objective Function for Image Restoration with Proximity Splitting Terms
6.2.2 Variational Conditions for Optimization of the Objective Function
6.2.3 Algorithm Derivation
6.3 Convergence Analysis
6.3.1 Convergence Proof
6.3.2 Convergence Rate Analysis
6.4 Further Discussion and Extension of the Primal-Dual Splitting Method
6.4.1 Relation to Parallel Linear Alternating Direction Method of Multipliers
6.4.2 Further Extensions of the Parallel Primal-Dual Splitting Method
6.5 Application of PPDS to TGV/Shearlet Compound Regularized Image Restoration
6.6 Experimental Results
6.6.1 Image Deblurring Experiment
6.6.2 Image Inpainting Experiment
6.6.3 Image Compressed Sensing Experiments
6.6.4 Experiments on the Validity of Pixel Interval Constraints
References
Appendix A
Appendix B
Uncited References
Index