Image Recovery: Theory and Application

Image Recovery: Theory and Application

1st Edition - March 2, 1987

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  • Editor: Henry Stark
  • eBook ISBN: 9780323145978

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Image Recovery: Theory and Application focuses on signal recovery and synthesis problems. This book discusses the concepts of image recovery, including regularization, the projection theorem, and the pseudoinverse operator. Comprised of 13 chapters, this volume begins with a review of the basic properties of linear vector spaces and associated operators, followed by a discussion on the Gerchberg-Papoulis algorithm. It then explores image restoration and the basic mathematical theory in image restoration problems. The reader is also introduced to the problem of obtaining artifact-free computed tomographic reconstruction. Other chapters consider the importance of Bayesian approach in the context of medical imaging. In addition, the book discusses the linear programming method, which is particularly important for images with large number of pixels with zero value. Such images are usually found in medical imaging, microscopy, electron microscopy, and astronomy. This book can be a valuable resource to materials scientists, engineers, computed tomography technologists, and astronomers.

Table of Contents

  • Preface


    Chapter 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


    Chapter 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


    Chapter 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


    Chapter 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


    Chapter 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


    Chapter 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


    Chapter 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


    Chapter 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


    Chapter 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


    Chapter 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

    10.8 Concluding Remarks


    Chapter 11 Applications of Convex Projection Theory to Image Recovery in Tomography and Related Areas

    11.1 Introduction

    11.2 Demonstration of the Method of Projections onto Convex Sets: Spectral Extrapolation of Images

    11.3 Review of the Direct Fourier Method

    11.4 POCS-DFM Algorithm

    11.5 A Case Study: Reconstruction of a Thorax Cross Section from Angularly Limited X-Ray Projection Data

    11.6 Applications to Related Areas

    11.7 Summary


    Chapter 12 Image Synthesis: Discovery Instead of Recovery

    12.1 Introduction

    12.2 General Formulation of the Problem

    12.3 Diffraction-Limited Systems

    12.4 Binary Images through Diffraction-Limited Systems

    12.5 Binary Images of Binary Masks

    12.6 Complex Masks for Coherent Imaging Systems

    12.7 Summary


    Chapter 13 The Role of Analyticity in Image Recovery

    13.1 Introduction

    13.2 Multidimensional Bandlimited Functions

    13.3 Consequences of Analyticity

    13.4 Image Reconstruction from Limited Spectral Data

    13.5 Image Reconstruction from Zeros

    13.6 Image Reconstruction from Polynomial Representation of Two-Dimensional Bandlimited Functions

    13.7 Analyticity and Phase Retrieval

    13.8 Analyticity and Phase Unwrapping

    13.9 Summary and Conclusions



Product details

  • No. of pages: 564
  • Language: English
  • Copyright: © Academic Press 1987
  • Published: March 2, 1987
  • Imprint: Academic Press
  • eBook ISBN: 9780323145978

About the Editor

Henry Stark

Affiliations and Expertise

Illinois Institute of Technology

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