Image Recovery: Theory and Application

PrefaceAcknowledgmentsChapter 1 Signal Restoration, Functional Analysis, and Fredholm Integral Equations of the First Kind 1.1 Introduction 1.2 Hilbert Spaces and Linear Operators 1.3 Existence of Solutions 1.4 Least-Squares Solutions and the Operator Pseudoinverse 1.5 Regularization 1.6 The Truncated SVD Expansion and Filtering 1.7 The Iterative Algorithm of Landweber 1.8 Alternating Orthogonal Projections 1.9 Regularized Iterative Algorithms 1.10 Moment Discretization 1.11 Summary and Conclusions ReferencesChapter 2 Mathematical Theory of Image Restoration by the Method of Convex Projections 2.1 Introduction 2.2 Some Properties of Convex Sets in Hilbert Space 2.3 Nonexpansive Maps and Their Fixed Points-Basic Theorems 2.4 Iterative Techniques for Image Restoration in a Hubert Space Setting 2.5 Useful Projections 2.6 Summary and New Developments ReferencesChapter 3 Bayesian and Related Methods in Image Reconstruction from Incomplete Data 3.1 Introduction 3.2 Measurement Space-Null Space 3.3 Deterministic Solutions 3.4 The Bayesian Approach 3.5 Use of Other Kinds of Prior Knowledge 3.6 MAP Solutions 3.7 FAIR-Fit and Iterative Reconstruction 3.8 Comparison of MAP and FAIR Results 3.9 A Generalized Bayesian Method 3.10 Discussion 3.11 Summary ReferencesChapter 4 Image Restoration Using Linear Programming 4.1 Image Restoration 4.2 Numerical Example of the Matrix Diagonalization of H 4.3 Linear Programming 4.4 Norms of the Error 4.5 Numerical Example of Minimum L1 Norm Method 4.6 Computation Considerations 4.7 Spatial Resolution 4.8 Results 4.9 Summary and Conclusions ReferencesChapter 5 The Principle of Maximum Entropy in Image Recovery 5.1 Introduction 5.2 Frieden's Approach 5.3 Burch, Gull, and Skilling's Approach 5.4 A Differential Equation Approach to Maximum Entropy Image Restoration 5.5 Conclusion ReferencesChapter 6 The Unique Reconstruction of Multidimensional Sequences from Fourier Transform Magnitude or Phase 6.1 Introduction 6.2 Fourier Synthesis from Partial Information 6.3 The Algebra of Polynomials in Two or More Variables 6.4 The Magnitude Retrieval Problem 6.5 The Phase Retrieval Problem 6.6 Summary and Other Problems ReferencesChapter 7 Phase Retrieval and Image Reconstruction for Astronomy 7.1 Introduction 7.2 Uniqueness of Phase Retrieval from Modulus Data 7.3 Algorithms for Phase Retrieval from Modulus 7.4 Iterative Transform Algorithm 7.5 Solutions Specific to Measurement Techniques 7.6 Conclusions ReferencesChapter 8 Restoration from Phase and Magnitude by Generalized Projections 8.1 Introduction 8.2 The Gerchberg-Saxton and Related Algorithms 8.3 The Method of Projections onto Convex Sets 8.4 Application of POCS to the Problem of Restoration from Phase 8.5 Computer Simulations of Restoration from Phase 8.6 The Method of Generalized Projections 8.7 Signal Recovery from Magnitude by Generalized Projections 8.8 Computer Simulations of Restoration from Magnitude 8.9 Conclusion ReferencesChapter 9 Image Reconstruction from Limited Data: Theory and Applications in Computerized Tomography 9.1 Introduction 9.2 Review of Image Reconstruction 9.3 An Inner Product Framework for Image Reconstruction 9.4 Applications of the Inner Product Framework 9.5 Image Reconstruction from Incomplete Data 9.6 Conclusions ReferencesChapter 10 Computer-Assisted Diffraction Tomography 10.1 Introduction 10.2 Transformations of the Wave Equation 10.3 Fourier Slice in Diffraction Tomography 10.4 Reconstruction Algorithms 10.5 Reconstruction from Limited Angular Data 10.6 Phase Determination 10.7 Experimental Results and Comparison of Born and Rytov Methods 10.