Digital Image Restoration, Softcover reprint of the original 1st ed. 1991 Coll. Springer Series in Information Sciences, Vol. 23

The field of image restoration is concerned with the estimation of uncorrupted im­ ages from noisy, blurred ones. These blurs might be caused by optical distortions, object motion during imaging, or atmospheric turbulence. In many scientific and en­ gineering applications, such as aerial imaging, remote sensing, electron microscopy, and medical imaging, there is active or potential work in image restoration. The purpose of this book is to provide in-depth treatment of some recent ad­ vances in the field of image restoration. A survey of the field is provided in the introduction. Recent research results are presented, regarding the formulation of the restoration problem as a convex programming problem, the implementation of restoration algorithms using artificial neural networks, the derivation of non­ stationary image models (compound random fields) and their application to image estimation and restoration, the development of algorithms for the simultaneous image and blur parameter identification and restoration, and the development of algorithms for restoring scanned photographic images. Special attention is directed to issues of numerical implementation. A large number of pictures demonstrate the performance of the restoration approaches. This book provides a clear understanding of the past achievements, a detailed description of the very important recent developments and the limitations of existing approaches, in the rapidly growing field of image restoration. It will be useful both as a reference book for working scientists and engineers and as a supplementary textbook in courses on image processing.

1. Introduction.- 1.1 The Digital Image Restoration Problem.- 1.2 Degradation Models.- 1.3 Image Models.- 1.4 Ill-Posed Problems and Regularization Approaches.- 1.4.1 Ill-Posed Problems.- 1.4.2 Regularization Approaches.- 1.5 Overview of Image Restoration Approaches.- 1.5.1 Deterministic Restoration Algorithms.- 1.5.2 Stochastic Algorithms.- 1.6 Discussion.- References.- 2. A Dual Approach to Signal Restoration.- 2.1 Background.- 2.2 Application of Convex Programming to Image Restoration.- 2.2.1 The Unified Cost Functional.- 2.2.2 Signal Constraints.- 2.2.3 Noise Constraints.- 2.2.4 Restatement of the Image Restoration Problem.- 2.3 The Dual Approach to Signal Restoration.- 2.3.1 The Primal and Dual Approaches.- 2.3.2 Generating Dual Functionals and Signal Models.- 2.3.3 Signal Models for Common Optimization Criteria.- 2.3.4 Existence and Uniqueness of the Optimally Restored Signal.- 2.4 Numerical Implementation and Results.- 2.4.1 Application of Optimization Algorithms to the Dual Problem.- 2.4.2 Comparison of Restoration Methods.- 2.4.3 Behavior of Optimization Procedures when a Feasible Solution does not Exist.- 2.5 Cost Functionals for Sequential Restoration.- 2.5.1 The Prior Estimate Consistency Condition.- 2.5.2 The Subsequent Estimate Consistency Condition.- 2.5.3 Modifications to the Entropy and Cross Entropy Functionals.- 2.6 Relationship Between the Original and Modified Entropy and Cross Entropy Functionals.- References.- 3. Hopfield-Type Neural Networks.- 3.1 Overview.- 3.2 Outline of the Chapter.- 3.3 The Hopfield-Type Associative Content Addressable Memory.- 3.3.1 Principles of Operation.- 3.3.2 The Methods of Projections onto Convex Sets and Generalized Projections.- 3.3.3 GP Formulation of the Binary Hopfield ACAM.- 3.3.4 POCS Formulation of a Continuous Hopfield ACAM.- 3.3.5 The Hopfield-Type Classifier: The ACAM Followed by a Perceptron.- 3.4 Image Restoration Using a Hopfield-Type Neural Network.- 3.4.1 Energy Reduction Property and Stable States.- 3.4.2 Network Model for Image Restoration.- 3.4.3 Remarks on the Restoration Network.- 3.4.4 Learning the Constraint.- 3.4.5 Simulation Results.- 3.5 Summary and Conclusion.- 3.A Appendices.- 3.A.1 Orthogonalization Learning Rule.- 3.A.2 Projection Operator of Cs.- 3.A.3 Proof of the Monotonicity of ? (?).- 3.A.4 Derivation of (3.68).- References.- 4. Compound Gauss-Markov Models for Image Processing.- 4.1 Overview.- 4.2 Compound Markov Random Fields.- 4.2.1 Compound Gauss-Markov Random Fields.- 4.2.2 Doubly Stochastic Gaussian Random Fields.- 4.3 Joint MAP Estimator.- 4.3.1 Simulated Annealing Approach.- 4.3.2 Deterministic Search for the MAP Estimate.- 4.4 Parameter Identification and Simulation Results.- 4.4.1 Parameter Identification.- 4.4.2 Experimental Results.- 4.5 Texture Segmentation.- 4.5.1 Mathematical Models for Texture Segmentation.- 4.5.2 Supervised and Unsupervised Algorithms.- 4.5.3 Experimental Results.- 4.6 Conclusions.- References.- 5. Image Estimation Using 2D Noncausal Gauss-Markov Random Field Models.- 5.1 Preliminaries.- 5.2 Model Representation.- 5.2.1 The GMRF Model.- 5.2.2 The Compound GMRF Model.- 5.3 Estimation in GMRF Models.- 5.3.1. Coding Method.- 5.3.2 A Consistent Estimation Scheme.- 5.3.3 Maximum Likelihood Estimation of Parameters.- 5.3.4 Parameter Estimation for the Compound Model.- 5.4 Relaxation Algorithms for MAP Estimation.- 5.4.1 Image and Noise Models.- 5.4.2 Stochastic Relaxation.- 5.4.3 Quantization Effects.- 5.4.4 Experimental Results.- 5.5 GNC Algorithm for MAP Estimation of Images Modeled by Compound GMRF.- 5.5.1 Experimental Results.- 5.A Appendices.- 5.A.1 Proof of Theorem 1.- 5.A.2 Proof of Theorem 2.- References.- 6. Maximum Likelihood Identification and Restoration of Images Using the Expectation-Maximization Algorithm.- 6.1 Overview.- 6.2 Image and Blur Models.- 6.3 ML Parameter Identification.- 6.3.1 Formulation.- 6.3.2 Constraints on the Unknown Parameters.- 6.4 ML Identification via the EM Algorithm.- 6.4.1 The EM Algorithm in Review.- 6.4.2 Alternating Optimization of Cross Entropy.- 6.4.3 The EM Algorithm in the Linear Gaussian Case.- 6.5 The EM Iterations for the ML Estimation of ø.- 6.5.1 {x,y} as the Complete Data.- 6.5.2 {x,v} as the Complete Data.- 6.5.3 {Dx, v} as the Complete Data.- 6.5.4 Iterative Wiener Filtering.- 6.6 Modified Forms of the Proposed Algorithm.- 6.6.1 ML Estimation of øAR.- 6.6.2 Spatial Domain Iteration.- 6.6.3 Parameterization of the Image and Blur Models.- 6.7 Experimental Results.- 6.8 Conclusions.- 6.A Appendix: Detailed Derivation of Eqs. (6.43–45).- References.- 7. Nonhomogeneous Image Identification and Restoration Procedures.- 7.1 Image Modeling.- 7.2 Kalman-Type Filtering for Restoration.- 7.2.1 The 2D Kalman Filter for Image Restoration.- 7.2.2 The ROMKF.- 7.2.3 Comparison of the ROMKF with Kalman Filter Implementations.- 7.2.4 Concluding Comment.- 7.3 Parameter Identification.- 7.3.1 Literature Review.- 7.3.2 Identification of Model Parameters.- 7.3.3 Experimental Results.- 7.3.4 Summary.- 7.4 Adaptive Image Restoration.- 7.4.1 Literature Review.- 7.4.2 Adaptive Approaches.- 7.4.3 Experimental Results.- 7.4.4 Summary.- 7.5 Conclusion.- 7.A Appendix: The Kalman Filter I.- References.- 8. Restoration of Scanned Photographic Images.- 8.1 Motivation.- 8.2 Modeling Scanned Blurred Photographic Images.- 8.2.1 Linear Space-Invariant Blur Modeling.- 8.2.2 Effect of Photographic Film Characteristics.- 8.2.3 Scanner Characteristics and Noise.- 8.3 Restoration of Photographic Images: Theory.- 8.3.1 The Domain for Deconvolution.- 8.3.2 Deconvolution with Multiplicative Noise.- 8.3.3 Suboptimal Restoration in the Exposure Domain.- 8.4 Restoration of Photographic Images: Practice.- 8.4.1 Blur Identification.- 8.4.2 Estimation of Other Filter Parameters and Procedure.- 8.4.3 Limitations in Restoring Photographically Blurred Images.- 8.5 Results.- 8.6 Conclusion.- References.- Additional References.