Jonathan M. Blackledge, Loughborough University, UK

• About the AuthorForewordPrefaceAcknowledgementsNotation AlphabeticalGreekOperatorsGlossary Mathematical and StatisticalComputer ScienceOrganizational and StandardsIntroduction Imaging ScienceSignals and ImagesImage FormationImage InformationImage AnalysisDigital Image ProcessingFundamental ProblemsAbout this BookSummary of Important ResultsPart I: Mathematical and Computational Background Chapter 1: Vector Fields 1.1 Scalar Fields1.2 Vector Fields1.3 The Divergence Theorem1.4 Summary of Important ResultsChapter 2: 2D Fourier Theory 2.1 The 2D Complex Fourier Series2.2 The 2D Delta Function2.3 The 2D Fourier Transform2.4 Physical Representation2.5 The Spectrum2.6 Definitions and Notation2.7 Some Important Results2.8 Some Important Theorems2.9 Convolution and Correlation2.10 Convolution and Correlation Theorems2.11 Other Integral Transforms2.12 Discussion2.13 Summary of Important ResultsChapter 3: The 2D DFT, FFT and FIR Filter 3.1 The Discrete Fourier Transform3.2 The Sampling Theorem3.3 The Discrete Spectrum of a Digital Image3.4 The Fast Fourier Transform3.5 The Imaging Equation and Convolution in 2D3.6 The Finite Impulse Response Filter3.7 Origin of the Imaging Equation3.8 Summary of Important ResultsChapter 4: Field and Wave Equations 4.1 The Langevin Equation4.2 Maxwell’s Equations4.3 General Solution to Maxwell’s (Microscopic) Equations4.4 The Macroscopic Maxwell’s Equations4.5 EM Waves in a Homogeneous Medium4.6 EM Waves in an Inhomogeneous Medium4.7 Elastic Field Equations4.8 Inhomogeneous Elastic Wave Equation4.9 Acoustic Field Equations4.10 Discussion4.11 Summary of Important ResultsChapter 5: Green Functions 5.1 Overview5.2 Introduction to the Green Function5.3 The Time Independent Wave Operator5.4 Wavefields Generated by Sources5.5 Time Dependent Green Function5.6 Time Dependent Sources5.7 Green Function Solution to Maxwell’s Equation5.8 The Diffusion Equation5.9 Green Function Solution to the Diffusion Equation5.10 The Laplace and Poisson Equations5.11 Discussion5.12 Summary of Important ResultsProblems: Part IPart II: Imaging Systems Modelling Chapter 6: Scattering Theory 6.1 The Schrödinger and Helmholtz Equations6.2 Solution to the Helmholtz Equation6.3 Examples of Born Scattering6.4 Other Approximation Methods6.5 The Born Series6.6 Inverse Scattering6.7 Surface Scattering Theory6.8 Summary of Important ResultsChapter 7: Imaging of Layered Media 7.1 Pulse-Echo Imaging7.2 EM Imaging of a Layered Dielectric7.3 Acoustic Imaging of a Layered Material7.4 Side-band Systems and Demodulation7.5 Some Applications7.6 Case Study: Imaging the Ionosphere7.7 Case Study: Radar Plasma Screening7.8 Summary of Important ResultsChapter 8: Projection Tomography 8.1 Basic Principles8.2 Projection Tomography and Scattering Theory8.3 The Radon Transform8.4 Back-Projection PSF8.5 The Central Slice Theorem8.6 Numerical Methods8.7 The Hough Transform8.8 Non-separable Image Processing8.9 Summary of Important ResultsChapter 9: Diffraction Tomography 9.1 Diffraction Tomography using CW Fields9.2 Pulse Mode Diffraction Tomography9.3 The Diffraction Slice Theorem9.4 Quantitative Diffraction Tomography9.5 EM Diffraction Tomography9.6 Case Study: Simulation of an Ultrasonic B-Scan9.7 Summary of Important ResultsChapter 10: Synthetic Aperture Imaging 10.1 Synthetic Aperture Radar10.2 Principles of SAR10.3 Electromagnetic Scattering Model for SAR10.4 Case Study: The ‘Sea Spikes’ Problem10.5 Quantitative Imaging with SAR10.6 Synthetic Aperture Sonar10.7 Summary of Important ResultsChapter 11: Optical Image Formation 11.1 Optical Diffraction11.2 The Fourier Transforming Properties of a Lens11.3 Linear Systems11.4 Images of Lines and Edges11.5 Linearity of Optical Imaging Systems11.6 Coherent Image Formation11.7 Phase Contrast Imaging11.8 Incoherent Image Formation11.9 Coherent and Incoherent Optical Imaging11.10 Optical Beams11.11 The Paraxial Wave Equation11.12 Holographic Imaging11.13 Case Study: Digital Watermarking11.14 Summary of Important ResultsProblems: Part IIPart III: Digital Image Processing Methods Chapter 12: Image Restoration and Reconstruction 12.1 Introduction12.2 Image Restoration12.3 The Inverse Filter12.4 The Wiener Filter12.5 The Power Spectrum Equalization Filter12.6 The Matched Filter12.7 Maximum Entropy Deconvolution12.8 Constrained Deconvolution12.9 Phase Reconstruction and Phase Imaging12.10 Non-stationary Deconvolution12.11 Discussion12.12 Summary of Important ResultsChapter 13: Reconstruction of Band-limited Images 13.1 The Gerchberg-Papoulis Method13.2 Incorporation of a Priori Information13.3 Example Demonstration and Applications13.4 Error Reduction Algorithm13.5 Discussion13.6 Summary of Important ResultsChapter 14: Bayesian Estimation Methods 14.1 Introduction to Probability and Bayes Rule14.2 The Maximum Likelihood Filter14.3 The Maximum a Posteriori Filter14.4 Super Resolution using Bayesian Methods14.5 Summary of Important ResultsChapter 15: Image Enhancement 15.1 Basic Transforms15.2 Histogram Equalization15.3 Homomorphic Filtering15.4 Light Diffusion and the High Emphasis Filter15.5 Noise Reduction15.6 The Median Filter15.7 Summary of Important ResultsProblems: Part IIIPart IV: Pattern Recognition and Computer Vision Chapter 16: Segmentation and Edge Detection 16.1 Correlation and the Auto-covariance Function16.2 Thresholding16.3 Edge Detection16.4 Second Order Edge Detection16.5 The Marr-Hildreth Method16.6 Pixel Clustering16.7 Clustering Tools16.8 Hierarchical Data Structures16.9 Summary of Important ResultsChapter 17: Statistical Modelling and Analysis 17.1 Random Scattering Theory17.2 Statistical Modelling Methods17.3 Phase Distribution Analysis17.4 Fully Coherent Scattering Processes17.5 Statistical Moments17.6 Noise and Statistical Tests17.7 Texture Segmentation17.8 Summary of Important ResultsChapter 18: Fractal Images and Image Processing 18.1 Introduction18.2 Geometry and Dimension18.3 Fractal Curves and Fractal Signals18.4 Random Scaling Fractals and Texture18.5 Methods of Computing the Fractal Dimension18.6 The Fourier and Fractal Dimensions18.7 Other Dimensions and Higher Order Fractals18.8 The Information Dimension18.9 The Lyapunov Dimension18.10 Fractal Images and Mandelbrot Surfaces18.11 Generalized Random Scaling Fractal (RSF) Models18.12 Multi-Fractal Analysis18.13 Case Study: Fractional Light Diffusion18.14 Summary of Important ResultsChapter 19: Coding and Compression 19.1 The Reasons for Compression19.2 Lossless Coding Methods19.3 Lossy Coding Methods19.4 Fractal Image Compression19.5 Properties and Features19.6 Improved Fractal Compression19.7 Compression Conscious Operations19.8 Fractal Texture Maps19.9 Summary of Important ResultsProblems: Part IVSummaryAppendix A: Solutions to Problems Solutions to Problems: Part ISolutions to Part IISolutions to Problems: Part IIISolutions to Problems: Part IVAppendix B: Supplementary ProblemsAppendix C: Fourier Transform of a FractalAppendix D: I/O and Graphics Utilities Reading and Writing Images to and From a Named Data FileDisplaying a Digital ImageIndex

Jonathan M. Blackledge, Loughborough University, UK

• About the AuthorForewordPrefaceAcknowledgementsNotation AlphabeticalGreekOperatorsGlossary Mathematical and StatisticalComputer ScienceOrganizational and StandardsIntroduction Imaging ScienceSignals and ImagesImage FormationImage InformationImage AnalysisDigital Image ProcessingFundamental ProblemsAbout this BookSummary of Important ResultsPart I: Mathematical and Computational Background Chapter 1: Vector Fields 1.1 Scalar Fields1.2 Vector Fields1.3 The Divergence Theorem1.4 Summary of Important ResultsChapter 2: 2D Fourier Theory 2.1 The 2D Complex Fourier Series2.2 The 2D Delta Function2.3 The 2D Fourier Transform2.4 Physical Representation2.5 The Spectrum2.6 Definitions and Notation2.7 Some Important Results2.8 Some Important Theorems2.9 Convolution and Correlation2.10 Convolution and Correlation Theorems2.11 Other Integral Transforms2.12 Discussion2.13 Summary of Important ResultsChapter 3: The 2D DFT, FFT and FIR Filter 3.1 The Discrete Fourier Transform3.2 The Sampling Theorem3.3 The Discrete Spectrum of a Digital Image3.4 The Fast Fourier Transform3.5 The Imaging Equation and Convolution in 2D3.6 The Finite Impulse Response Filter3.7 Origin of the Imaging Equation3.8 Summary of Important ResultsChapter 4: Field and Wave Equations 4.1 The Langevin Equation4.2 Maxwell’s Equations4.3 General Solution to Maxwell’s (Microscopic) Equations4.4 The Macroscopic Maxwell’s Equations4.5 EM Waves in a Homogeneous Medium4.6 EM Waves in an Inhomogeneous Medium4.7 Elastic Field Equations4.8 Inhomogeneous Elastic Wave Equation4.9 Acoustic Field Equations4.10 Discussion4.11 Summary of Important ResultsChapter 5: Green Functions 5.1 Overview5.2 Introduction to the Green Function5.3 The Time Independent Wave Operator5.4 Wavefields Generated by Sources5.5 Time Dependent Green Function5.6 Time Dependent Sources5.7 Green Function Solution to Maxwell’s Equation5.8 The Diffusion Equation5.9 Green Function Solution to the Diffusion Equation5.10 The Laplace and Poisson Equations5.11 Discussion5.12 Summary of Important ResultsProblems: Part IPart II: Imaging Systems Modelling Chapter 6: Scattering Theory 6.1 The Schrödinger and Helmholtz Equations6.2 Solution to the Helmholtz Equation6.3 Examples of Born Scattering6.4 Other Approximation Methods6.5 The Born Series6.6 Inverse Scattering6.7 Surface Scattering Theory6.8 Summary of Important ResultsChapter 7: Imaging of Layered Media 7.1 Pulse-Echo Imaging7.2 EM Imaging of a Layered Dielectric7.3 Acoustic Imaging of a Layered Material7.4 Side-band Systems and Demodulation7.5 Some Applications7.6 Case Study: Imaging the Ionosphere7.7 Case Study: Radar Plasma Screening7.8 Summary of Important ResultsChapter 8: Projection Tomography 8.1 Basic Principles8.2 Projection Tomography and Scattering Theory8.3 The Radon Transform8.4 Back-Projection PSF8.5 The Central Slice Theorem8.6 Numerical Methods8.7 The Hough Transform8.8 Non-separable Image Processing8.9 Summary of Important ResultsChapter 9: Diffraction Tomography 9.1 Diffraction Tomography using CW Fields9.2 Pulse Mode Diffraction Tomography9.3 The Diffraction Slice Theorem9.4 Quantitative Diffraction Tomography9.5 EM Diffraction Tomography9.6 Case Study: Simulation of an Ultrasonic B-Scan9.7 Summary of Important ResultsChapter 10: Synthetic Aperture Imaging 10.1 Synthetic Aperture Radar10.2 Principles of SAR10.3 Electromagnetic Scattering Model for SAR10.4 Case Study: The ‘Sea Spikes’ Problem10.5 Quantitative Imaging with SAR10.6 Synthetic Aperture Sonar10.7 Summary of Important ResultsChapter 11: Optical Image Formation 11.1 Optical Diffraction11.2 The Fourier Transforming Properties of a Lens11.3 Linear Systems11.4 Images of Lines and Edges11.5 Linearity of Optical Imaging Systems11.6 Coherent Image Formation11.7 Phase Contrast Imaging11.8 Incoherent Image Formation11.9 Coherent and Incoherent Optical Imaging11.10 Optical Beams11.11 The Paraxial Wave Equation11.12 Holographic Imaging11.13 Case Study: Digital Watermarking11.14 Summary of Important ResultsProblems: Part IIPart III: Digital Image Processing Methods Chapter 12: Image Restoration and Reconstruction 12.1 Introduction12.2 Image Restoration12.3 The Inverse Filter12.4 The Wiener Filter12.5 The Power Spectrum Equalization Filter12.6 The Matched Filter12.7 Maximum Entropy Deconvolution12.8 Constrained Deconvolution12.9 Phase Reconstruction and Phase Imaging12.10 Non-stationary Deconvolution12.11 Discussion12.12 Summary of Important ResultsChapter 13: Reconstruction of Band-limited Images 13.1 The Gerchberg-Papoulis Method13.2 Incorporation of a Priori Information13.3 Example Demonstration and Applications13.4 Error Reduction Algorithm13.5 Discussion13.6 Summary of Important ResultsChapter 14: Bayesian Estimation Methods 14.1 Introduction to Probability and Bayes Rule14.2 The Maximum Likelihood Filter14.3 The Maximum a Posteriori Filter14.4 Super Resolution using Bayesian Methods14.5 Summary of Important ResultsChapter 15: Image Enhancement 15.1 Basic Transforms15.2 Histogram Equalization15.3 Homomorphic Filtering15.4 Light Diffusion and the High Emphasis Filter15.5 Noise Reduction15.6 The Median Filter15.7 Summary of Important ResultsProblems: Part IIIPart IV: Pattern Recognition and Computer Vision Chapter 16: Segmentation and Edge Detection 16.1 Correlation and the Auto-covariance Function16.2 Thresholding16.3 Edge Detection16.4 Second Order Edge Detection16.5 The Marr-Hildreth Method16.6 Pixel Clustering16.7 Clustering Tools16.8 Hierarchical Data Structures16.9 Summary of Important ResultsChapter 17: Statistical Modelling and Analysis 17.1 Random Scattering Theory17.2 Statistical Modelling Methods17.3 Phase Distribution Analysis17.4 Fully Coherent Scattering Processes17.5 Statistical Moments17.6 Noise and Statistical Tests17.7 Texture Segmentation17.8 Summary of Important ResultsChapter 18: Fractal Images and Image Processing 18.1 Introduction18.2 Geometry and Dimension18.3 Fractal Curves and Fractal Signals18.4 Random Scaling Fractals and Texture18.5 Methods of Computing the Fractal Dimension18.6 The Fourier and Fractal Dimensions18.7 Other Dimensions and Higher Order Fractals18.8 The Information Dimension18.9 The Lyapunov Dimension18.10 Fractal Images and Mandelbrot Surfaces18.11 Generalized Random Scaling Fractal (RSF) Models18.12 Multi-Fractal Analysis18.13 Case Study: Fractional Light Diffusion18.14 Summary of Important ResultsChapter 19: Coding and Compression 19.1 The Reasons for Compression19.2 Lossless Coding Methods19.3 Lossy Coding Methods19.4 Fractal Image Compression19.5 Properties and Features19.6 Improved Fractal Compression19.7 Compression Conscious Operations19.8 Fractal Texture Maps19.9 Summary of Important ResultsProblems: Part IVSummaryAppendix A: Solutions to Problems Solutions to Problems: Part ISolutions to Part IISolutions to Problems: Part IIISolutions to Problems: Part IVAppendix B: Supplementary ProblemsAppendix C: Fourier Transform of a FractalAppendix D: I/O and Graphics Utilities Reading and Writing Images to and From a Named Data FileDisplaying a Digital ImageIndex

Jonathan M. Blackledge, Loughborough University, UK

• About the AuthorForewordPrefaceAcknowledgementsNotation AlphabeticalGreekOperatorsGlossary Mathematical and StatisticalComputer ScienceOrganizational and StandardsIntroduction Imaging ScienceSignals and ImagesImage FormationImage InformationImage AnalysisDigital Image ProcessingFundamental ProblemsAbout this BookSummary of Important ResultsPart I: Mathematical and Computational Background Chapter 1: Vector Fields 1.1 Scalar Fields1.2 Vector Fields1.3 The Divergence Theorem1.4 Summary of Important ResultsChapter 2: 2D Fourier Theory 2.1 The 2D Complex Fourier Series2.2 The 2D Delta Function2.3 The 2D Fourier Transform2.4 Physical Representation2.5 The Spectrum2.6 Definitions and Notation2.7 Some Important Results2.8 Some Important Theorems2.9 Convolution and Correlation2.10 Convolution and Correlation Theorems2.11 Other Integral Transforms2.12 Discussion2.13 Summary of Important ResultsChapter 3: The 2D DFT, FFT and FIR Filter 3.1 The Discrete Fourier Transform3.2 The Sampling Theorem3.3 The Discrete Spectrum of a Digital Image3.4 The Fast Fourier Transform3.5 The Imaging Equation and Convolution in 2D3.6 The Finite Impulse Response Filter3.7 Origin of the Imaging Equation3.8 Summary of Important ResultsChapter 4: Field and Wave Equations 4.1 The Langevin Equation4.2 Maxwell’s Equations4.3 General Solution to Maxwell’s (Microscopic) Equations4.4 The Macroscopic Maxwell’s Equations4.5 EM Waves in a Homogeneous Medium4.6 EM Waves in an Inhomogeneous Medium4.7 Elastic Field Equations4.8 Inhomogeneous Elastic Wave Equation4.9 Acoustic Field Equations4.10 Discussion4.11 Summary of Important ResultsChapter 5: Green Functions 5.1 Overview5.2 Introduction to the Green Function5.3 The Time Independent Wave Operator5.4 Wavefields Generated by Sources5.5 Time Dependent Green Function5.6 Time Dependent Sources5.7 Green Function Solution to Maxwell’s Equation5.8 The Diffusion Equation5.9 Green Function Solution to the Diffusion Equation5.10 The Laplace and Poisson Equations5.11 Discussion5.12 Summary of Important ResultsProblems: Part IPart II: Imaging Systems Modelling Chapter 6: Scattering Theory 6.1 The Schrödinger and Helmholtz Equations6.2 Solution to the Helmholtz Equation6.3 Examples of Born Scattering6.4 Other Approximation Methods6.5 The Born Series6.6 Inverse Scattering6.7 Surface Scattering Theory6.8 Summary of Important ResultsChapter 7: Imaging of Layered Media 7.1 Pulse-Echo Imaging7.2 EM Imaging of a Layered Dielectric7.3 Acoustic Imaging of a Layered Material7.4 Side-band Systems and Demodulation7.5 Some Applications7.6 Case Study: Imaging the Ionosphere7.7 Case Study: Radar Plasma Screening7.8 Summary of Important ResultsChapter 8: Projection Tomography 8.1 Basic Principles8.2 Projection Tomography and Scattering Theory8.3 The Radon Transform8.4 Back-Projection PSF8.5 The Central Slice Theorem8.6 Numerical Methods8.7 The Hough Transform8.8 Non-separable Image Processing8.9 Summary of Important ResultsChapter 9: Diffraction Tomography 9.1 Diffraction Tomography using CW Fields9.2 Pulse Mode Diffraction Tomography9.3 The Diffraction Slice Theorem9.4 Quantitative Diffraction Tomography9.5 EM Diffraction Tomography9.6 Case Study: Simulation of an Ultrasonic B-Scan9.7 Summary of Important ResultsChapter 10: Synthetic Aperture Imaging 10.1 Synthetic Aperture Radar10.2 Principles of SAR10.3 Electromagnetic Scattering Model for SAR10.4 Case Study: The ‘Sea Spikes’ Problem10.5 Quantitative Imaging with SAR10.6 Synthetic Aperture Sonar10.7 Summary of Important ResultsChapter 11: Optical Image Formation 11.1 Optical Diffraction11.2 The Fourier Transforming Properties of a Lens11.3 Linear Systems11.4 Images of Lines and Edges11.5 Linearity of Optical Imaging Systems11.6 Coherent Image Formation11.7 Phase Contrast Imaging11.8 Incoherent Image Formation11.9 Coherent and Incoherent Optical Imaging11.10 Optical Beams11.11 The Paraxial Wave Equation11.12 Holographic Imaging11.13 Case Study: Digital Watermarking11.14 Summary of Important ResultsProblems: Part IIPart III: Digital Image Processing Methods Chapter 12: Image Restoration and Reconstruction 12.1 Introduction12.2 Image Restoration12.3 The Inverse Filter12.4 The Wiener Filter12.5 The Power Spectrum Equalization Filter12.6 The Matched Filter12.7 Maximum Entropy Deconvolution12.8 Constrained Deconvolution12.9 Phase Reconstruction and Phase Imaging12.10 Non-stationary Deconvolution12.11 Discussion12.12 Summary of Important ResultsChapter 13: Reconstruction of Band-limited Images 13.1 The Gerchberg-Papoulis Method13.2 Incorporation of a Priori Information13.3 Example Demonstration and Applications13.4 Error Reduction Algorithm13.5 Discussion13.6 Summary of Important ResultsChapter 14: Bayesian Estimation Methods 14.1 Introduction to Probability and Bayes Rule14.2 The Maximum Likelihood Filter14.3 The Maximum a Posteriori Filter14.4 Super Resolution using Bayesian Methods14.5 Summary of Important ResultsChapter 15: Image Enhancement 15.1 Basic Transforms15.2 Histogram Equalization15.3 Homomorphic Filtering15.4 Light Diffusion and the High Emphasis Filter15.5 Noise Reduction15.6 The Median Filter15.7 Summary of Important ResultsProblems: Part IIIPart IV: Pattern Recognition and Computer Vision Chapter 16: Segmentation and Edge Detection 16.1 Correlation and the Auto-covariance Function16.2 Thresholding16.3 Edge Detection16.4 Second Order Edge Detection16.5 The Marr-Hildreth Method16.6 Pixel Clustering16.7 Clustering Tools16.8 Hierarchical Data Structures16.9 Summary of Important ResultsChapter 17: Statistical Modelling and Analysis 17.1 Random Scattering Theory17.2 Statistical Modelling Methods17.3 Phase Distribution Analysis17.4 Fully Coherent Scattering Processes17.5 Statistical Moments17.6 Noise and Statistical Tests17.7 Texture Segmentation17.8 Summary of Important ResultsChapter 18: Fractal Images and Image Processing 18.1 Introduction18.2 Geometry and Dimension18.3 Fractal Curves and Fractal Signals18.4 Random Scaling Fractals and Texture18.5 Methods of Computing the Fractal Dimension18.6 The Fourier and Fractal Dimensions18.7 Other Dimensions and Higher Order Fractals18.8 The Information Dimension18.9 The Lyapunov Dimension18.10 Fractal Images and Mandelbrot Surfaces18.11 Generalized Random Scaling Fractal (RSF) Models18.12 Multi-Fractal Analysis18.13 Case Study: Fractional Light Diffusion18.14 Summary of Important ResultsChapter 19: Coding and Compression 19.1 The Reasons for Compression19.2 Lossless Coding Methods19.3 Lossy Coding Methods19.4 Fractal Image Compression19.5 Properties and Features19.6 Improved Fractal Compression19.7 Compression Conscious Operations19.8 Fractal Texture Maps19.9 Summary of Important ResultsProblems: Part IVSummaryAppendix A: Solutions to Problems Solutions to Problems: Part ISolutions to Part IISolutions to Problems: Part IIISolutions to Problems: Part IVAppendix B: Supplementary ProblemsAppendix C: Fourier Transform of a FractalAppendix D: I/O and Graphics Utilities Reading and Writing Images to and From a Named Data FileDisplaying a Digital ImageIndex