This authoritative text (the second part of a complete MSc course) provides mathematical methods required to describe images, image formation and different imaging systems, coupled with the principle techniques used for processing digital images. It is based on a course for postgraduates reading physics, electronic engineering, telecommunications engineering, information technology and computer science. This book relates the methods of processing and interpreting digital images to the ‘physics’ of imaging systems. Case studies reinforce the methods discussed, with examples of current research themes.

Table of Contents:
About the Author Foreword Preface Acknowledgements Notation Alphabetical Greek Operators Glossary Mathematical and Statistical Computer Science Organizational and Standards Introduction Imaging Science Signals and Images Image Formation Image Information Image Analysis Digital Image Processing Fundamental Problems About this Book Summary of Important Results Part I: Mathematical and Computational Background Chapter 1: Vector Fields 1.1 Scalar Fields 1.2 Vector Fields 1.3 The Divergence Theorem 1.4 Summary of Important Results Chapter 2: 2D Fourier Theory 2.1 The 2D Complex Fourier Series 2.2 The 2D Delta Function 2.3 The 2D Fourier Transform 2.4 Physical Representation 2.5 The Spectrum 2.6 Definitions and Notation 2.7 Some Important Results 2.8 Some Important Theorems 2.9 Convolution and Correlation 2.10 Convolution and Correlation Theorems 2.11 Other Integral Transforms 2.12 Discussion 2.13 Summary of Important Results Chapter 3: The 2D DFT, FFT and FIR Filter 3.1 The Discrete Fourier Transform 3.2 The Sampling Theorem 3.3 The Discrete Spectrum of a Digital Image 3.4 The Fast Fourier Transform 3.5 The Imaging Equation and Convolution in 2D 3.6 The Finite Impulse Response Filter 3.7 Origin of the Imaging Equation 3.8 Summary of Important Results Chapter 4: Field and Wave Equations 4.1 The Langevin Equation 4.2 Maxwell’s Equations 4.3 General Solution to Maxwell’s (Microscopic) Equations 4.4 The Macroscopic Maxwell’s Equations 4.5 EM Waves in a Homogeneous Medium 4.6 EM Waves in an Inhomogeneous Medium 4.7 Elastic Field Equations 4.8 Inhomogeneous Elastic Wave Equation 4.9 Acoustic Field Equations 4.10 Discussion 4.11 Summary of Important Results Chapter 5: Green Functions 5.1 Overview 5.2 Introduction to the Green Function 5.3 The Time Independent Wave Operator 5.4 Wavefields Generated by Sources 5.5 Time Dependent Green Function 5.6 Time Dependent Sources 5.7 Green Function Solution to Maxwell’s Equation 5.8 The Diffusion Equation 5.9 Green Function Solution to the Diffusion Equation 5.10 The Laplace and Poisson Equations 5.11 Discussion 5.12 Summary of Important Results Problems: Part I Part II: Imaging Systems Modelling Chapter 6: Scattering Theory 6.1 The Schrödinger and Helmholtz Equations 6.2 Solution to the Helmholtz Equation 6.3 Examples of Born Scattering 6.4 Other Approximation Methods 6.5 The Born Series 6.6 Inverse Scattering 6.7 Surface Scattering Theory 6.8 Summary of Important Results Chapter 7: Imaging of Layered Media 7.1 Pulse-Echo Imaging 7.2 EM Imaging of a Layered Dielectric 7.3 Acoustic Imaging of a Layered Material 7.4 Side-band Systems and Demodulation 7.5 Some Applications 7.6 Case Study: Imaging the Ionosphere 7.7 Case Study: Radar Plasma Screening 7.8 Summary of Important Results Chapter 8: Projection Tomography 8.1 Basic Principles 8.2 Projection Tomography and Scattering Theory 8.3 The Radon Transform 8.4 Back-Projection PSF 8.5 The Central Slice Theorem 8.6 Numerical Methods 8.7 The Hough Transform 8.8 Non-separable Image Processing 8.9 Summary of Important Results Chapter 9: Diffraction Tomography 9.1 Diffraction Tomography using CW Fields 9.2 Pulse Mode Diffraction Tomography 9.3 The Diffraction Slice Theorem 9.4 Quantitative Diffraction Tomography 9.5 EM Diffraction Tomography 9.6 Case Study: Simulation of an Ultrasonic B-Scan 9.7 Summary of Important Results Chapter 10: Synthetic Aperture Imaging 10.1 Synthetic Aperture Radar 10.2 Principles of SAR 10.3 Electromagnetic Scattering Model for SAR 10.4 Case Study: The ‘Sea Spikes’ Problem 10.5 Quantitative Imaging with SAR 10.6 Synthetic Aperture Sonar 10.7 Summary of Important Results Chapter 11: Optical Image Formation 11.1 Optical Diffraction 11.2 The Fourier Transforming Properties of a Lens 11.3 Linear Systems 11.4 Images of Lines and Edges 11.5 Linearity of Optical Imaging Systems 11.6 Coherent Image Formation 11.7 Phase Contrast Imaging 11.8 Incoherent Image Formation 11.9 Coherent and Incoherent Optical Imaging 11.10 Optical Beams 11.11 The Paraxial Wave Equation 11.12 Holographic Imaging 11.13 Case Study: Digital Watermarking 11.14 Summary of Important Results Problems: Part II Part III: Digital Image Processing Methods Chapter 12: Image Restoration and Reconstruction 12.1 Introduction 12.2 Image Restoration 12.3 The Inverse Filter 12.4 The Wiener Filter 12.5 The Power Spectrum Equalization Filter 12.6 The Matched Filter 12.7 Maximum Entropy Deconvolution 12.8 Constrained Deconvolution 12.9 Phase Reconstruction and Phase Imaging 12.10 Non-stationary Deconvolution 12.11 Discussion 12.12 Summary of Important Results Chapter 13: Reconstruction of Band-limited Images 13.1 The Gerchberg-Papoulis Method 13.2 Incorporation of a Priori Information 13.3 Example Demonstration and Applications 13.4 Error Reduction Algorithm 13.5 Discussion 13.6 Summary of Important Results Chapter 14: Bayesian Estimation Methods 14.1 Introduction to Probability and Bayes Rule 14.2 The Maximum Likelihood Filter 14.3 The Maximum a Posteriori Filter 14.4 Super Resolution using Bayesian Methods 14.5 Summary of Important Results Chapter 15: Image Enhancement 15.1 Basic Transforms 15.2 Histogram Equalization 15.3 Homomorphic Filtering 15.4 Light Diffusion and the High Emphasis Filter 15.5 Noise Reduction 15.6 The Median Filter 15.7 Summary of Important Results Problems: Part III Part IV: Pattern Recognition and Computer Vision Chapter 16: Segmentation and Edge Detection 16.1 Correlation and the Auto-covariance Function 16.2 Thresholding 16.3 Edge Detection 16.4 Second Order Edge Detection 16.5 The Marr-Hildreth Method 16.6 Pixel Clustering 16.7 Clustering Tools 16.8 Hierarchical Data Structures 16.9 Summary of Important Results Chapter 17: Statistical Modelling and Analysis 17.1 Random Scattering Theory 17.2 Statistical Modelling Methods 17.3 Phase Distribution Analysis 17.4 Fully Coherent Scattering Processes 17.5 Statistical Moments 17.6 Noise and Statistical Tests 17.7 Texture Segmentation 17.8 Summary of Important Results Chapter 18: Fractal Images and Image Processing 18.1 Introduction 18.2 Geometry and Dimension 18.3 Fractal Curves and Fractal Signals 18.4 Random Scaling Fractals and Texture 18.5 Methods of Computing the Fractal Dimension 18.6 The Fourier and Fractal Dimensions 18.7 Other Dimensions and Higher Order Fractals 18.8 The Information Dimension 18.9 The Lyapunov Dimension 18.10 Fractal Images and Mandelbrot Surfaces 18.11 Generalized Random Scaling Fractal (RSF) Models 18.12 Multi-Fractal Analysis 18.13 Case Study: Fractional Light Diffusion 18.14 Summary of Important Results Chapter 19: Coding and Compression 19.1 The Reasons for Compression 19.2 Lossless Coding Methods 19.3 Lossy Coding Methods 19.4 Fractal Image Compression 19.5 Properties and Features 19.6 Improved Fractal Compression 19.7 Compression Conscious Operations 19.8 Fractal Texture Maps 19.9 Summary of Important Results Problems: Part IV Summary Appendix A: Solutions to Problems Solutions to Problems: Part I Solutions to Part II Solutions to Problems: Part III Solutions to Problems: Part IV Appendix B: Supplementary Problems Appendix C: Fourier Transform of a Fractal Appendix D: I/O and Graphics Utilities Reading and Writing Images to and From a Named Data File Displaying a Digital Image Index