This book is written forr graduate students and researchers in applied mathematics, computer science, electrical engineering, and other disciplines who are interested in problems in imaging and computer vision. It can be used as a reference by scientists with specific tasks in image processing, as well as by researchers with a general interest in finding out about the latest advances.

List of Figures

Preface

Introduction

1.1  Dawning of the Era of Imaging Sciences

 1.1.1 Image Acquisition

 1.1.2 Image Processing

 1.1.3 Image Interpretation and Visual Intelligence

1.2 Image Processing by Examples

 1.2.1 Image Contrast Enhancement

 1.2.2 Image Denoisirg

 1.2.3 Image Deblurring

 1.2.4 Image Inpainting

 1.2.5 Image Segmentation

1.3 An Overview of Methodologies in Image Processing

 1.3.1 Morphological Approach

 1.3.2 Fourier and Spectral Analysis

 1.3.3 Wavelet and Space-Scale Analysis

 1.3.4 Stochastic Modeling

 1.3.5 Variaticnal Methods

 1.3.6 Partial Differential Equations (PDEs)

 1.3.7 Different Approaches Are Intrinsically Interconnected

1.4 Organization of the Book

1.5 How to Read the Bcok

Some Modern Image Analysis Tools

2.1  Geometry of Curves and Surfaces

2.1.I Geometry of Curves

2.1.2 Geometry of Surfaces in Three Dimensions

2.1.3 Hausdorff Measures and Dimensions

2.2 Functions with Bounded Variations

2.2.1 Total Variatien as a Radon Measure

2.2.2 Basic Properties of BV Functions

2.2.3 The Co-Area Formula

2.3  Elements of Thermodynamics and Statistical Mechanics

2.3.1 Essentials of Thermodynamics

2.3.2 Entropy and Potentials

2.3.3 Statistical Mechanics of Ensembles

2.4 Bayesian Statistical Inference

2.4.1 Image Processing or Visual Perception as Inference

2.4.2 Bayesian Inference： Bias Due to Prior Knowledge

2.4.3 Bayesian Method in Image Processing

2.5 Linear and Nonlinear Filtering and Diffusion

2.5.1 Point Spreading and Markov Transition

2.5.2 Linear Filtering and Diffusion

2.5.3 Nonlinear Filtering and Diffusion

2.6 Wavelets and Multiresolution Analysis

2.6.1 Quest for New Image Analysis Tools

2.6.2 Early Edge Theory and Marr’s Wavelets

2.6.3 Windowed Frequency Analysis and Gabor Wavelets

2.6.4 Frequency-Window Coupling： Malvar-Wilson Wavelets

2.6.5 The Framework of Multiresolution Analysis (MRA)

2.6.6 Fast Image Analysis and Synthesis via Filter Banks

Image Modeling and Representation

3.1  Modeling and Representation： What, Why, and How

3.2 Deterministic Image Models

3.2.1 Images as Distributions (Generalized Functions)

3.2.2 Lp Images

3.2.3 Sobolev Images Hn(Ω)

3.2.4 BV Images

3.3 Wavelets and Multiscale Representation

3.3.1 Construction of 2-D Wavelets

3.3.2 Wavelet Responses to Typical Image Features

3.3.3 Besov Images and Sparse Wavelet Representation

3.4 Lattice and Random Field Representation

3.4.1 Natural Images of Mother Nature

3.4.2 Images as Ensembles and Distributions

3.4.3 Images as Gibbs’ Ensembles

3.4.4 Images as Markov Random Fields

3.4.5 Visual Filters and Filter Banks

3.4.6 Entropy-Based Learning of Image Patterns

3.5 Level-Set Representation

3.5.1 Classical Level Sets

3.5.2 Cumulative Level Sets

3.5.3 Level-Set Synthesis

3.5.4 An Example： Level Sets of Piecewise Constant Images

3.5.5 High Order Regularity of Level Sets

3.5.6 Statistics of Level Sets of Natural Images

3.6 The Mumford-Shah Free Boundary Image Model

3.6.1 Piecewise Constant 1-D Images： Analysis and Synthesis

3.6.2 Piecewise Smooth 1-D Images： First Order Representation

3.6.3 Piecewise Smooth 1-D Images： Poisson Representation

3.6.4 Piecewise Smooth 2-D Images

3.6.5 The Mumford-Shah Model

3.6.6 The Role of Special BV Images

Image Denoising

4.1  Noise： Origins, Physics, and Models

4.1.1 Origins and Physics of Noise

4.1.2 A Brief Overview of 1-D Stochastic Signals

4.1.3 Stochastic Models of Noises

4.1.4 Analog White Noises as Random Generalized Functions

4.1.5 Random Signals from Stochastic Differential Equations

4.1.6 2-D Stochastic Spatial Signals： Random Fields

4.2  Linear Denoising： Lowpass Filtering

4.2.1  Signal vs. Noise

4.2.2 Denoising via Linear Filters and Diffusion

4.3  Data-Driven Optimal Filtering： Wiener Filters

4.4 Wavelet Shrinkage Denoising

4.4.1 Shrinkage： Quasi-statistical Estimation of Singletons

4.4.2 Shrinkage： Variational Estimation of Singletons

4.4.3 Denoising via Shrinking Noisy Wavelet Components

4.4.4 Variational Denoising of Noisy Besov Images

4.5  Variational Denoising Based on BV Image Model

4.5.1 TV, Robust Statistics, and Median

4.5.2 The Role of TV and BV Image Model

4.5.3 Biased Iterated Median Filtering

4.5.4 Rudin, Osher, and Fatemi’s TV Denoising Model

4.5.5 Computational Approaches to TV Denoising

4.5.6 Duality for the TV Denoising Model

4.5.7 Solution Structures of the TV Denoising Model

4.6 Denoising via Nonlinear Diffusion and Scale-Space Theory

4.6.1 Perona and Malik’s Nonlinear Diffusion Model

4.6.2 Axiomatic Scale-Space Theory

4.7  Denoising Salt-and-Pepper Noise

4.8 Multichannel TV Denoising

4.8.1 Variational TV Denoising of Multichannel Images

4.8.2 Three Versions of TV[u]

Image Deblurring

5.1  Blur： Physical Origins and Mathematical Models

5.1.1 Physical Origins

5.1.2 Mathematical Models of Blurs

5.1.3 Linear vs. Nonlinear Blurs

5.2 Ill-posedness and Regularization

5.3 Deblurring with Wiener Filters

5.3.1 Intuition on Filter-Based Deblurring

5.3.2 Wiener Filtering

5.4 Deblurring of BV Images with Known PSF

5.4.1 The Variational Model

5.4.2 Existence and Uniqueness

5.4.3 Computation

5.5 Variational Blind Deblurring with Unknown PSF

5.5.1 Parametric Blind Deblurring

5.5.2 Parametric-Field-Based Blind Deblurring

5.5.3 Nonparametric Blind Deblurring

Image Inpainting

6.1 A Brief Review on Classical Interpolation Schemes

6.1.1 Polynomial Interpolation

6.1.2 Trigonometric Polynomial Interpolation

6.1.3 Spline Interpolation

6.1.4 Shannon’s Sampling Theorem

6.1.5 Radial Basis Functions and Thin-Plate Splines

6.2 Challenges and Guidelines for 2-D Image Inpainting

6.2.1 Main Challenges for Image Inpainting

6.2.2 General Guidelines for Image Inpainting

6.3 Inpainting of Sobolev Images： Green’s Formulae

6.4 Geometric Modeling of Curves and Images

6.4.1 Geometric Curve Models

6.4.2 2-, 3-Point Accumulative Energies, Length, and Curvature

6.4.3 Image Models via Functionalizing Curve Models

6.4.4 Image Models with Embedded Edge Models

6.5 Inpainting BV Images (via the TV Radon Measure)

6.5.1 Formulation of the TV Inpainting Model

6.5.2 Justification of TV Inpainting by Visual Perception

6.5.3 Computation of TV Inpainting

6.5.4 Digital Zooming Based on TV Inpainting

6.5.5 Edge-Based Image Coding via Inpainting

6.5.6 More Examples and Applications of TV Inpainting

6.6 Error Analysis for Image Inpainting

6.7 lnpainting Piecewise Smooth Images via Mumford and Shah

6.8 Image Inpainting via Euler’s Elasticas and Curvatures

6.8.1 Inpainting Based on the Elastica Image Model

6.8.2 Inpainting via Mumford-Shah-Euler Image Model

6.9 Inpainting of Meyer’s Texture

6.10 Image Inpainting with Missing Wavelet Coefficients

6.11 PDE Inpainting： Transport, Diffusion, and Navier-Stokes

6.11.1 Second Order Interpolation Models

6.11.2 A Third Order PDE Inpainting Model and Navier-Stokes

6.11.3 TV Inpainting Revisited： Anisotropic Diffusion

6.11.4 CDD Inpainting： Curvature Driven Diffusion

6.11.5 A Quasi-axiomatic Approach to Third Order Inpainting

6.12 Inpainting of Gibbs/Markov Random Fields

Image Segmentation

7.1  Synthetic Images： Monoids of Occlusive Preimages

7. I. I Introduction and Motivation

7.1.2 Monoids of Occlusive Preimages

7.1.3 Mimimal and Prime (or Atomic) Generators

7.2  Edges and Active Contours

7.2.1 Pixelwise Characterization of Edges： David Mart’s Edges

7.2.2 Edge-Regulated Data Models for Image Gray Values

7.2.3 Geometry-Regulated Prior Models for Edges

7.2.4 Active Contours： Combining Both Prior and Data Models

7.2.5 Curve Evolutions via Gradient Descent

7.2.6 F-Convergence Approximation of Active Contours

7.2.7 Region-Based Active Contours Driven by Gradients

7.2.8 Region-Based Active Contours Driven by Stochastic

Features

7.3  Geman and Geman’s Intensity-Edge Mixture Model

7.3.1 Topological Pixel Domains, Graphs, and Cliques

7.3.2 Edges as HiddenMarkov Random Fields

7.3.3 Intensities as Edge-Regulated Markov Random Fields

7.3.4 Gibbs’ Fields for Joint Bayesian Estimation of u and 1"

7.4 The Mumford-Shah Free-Boundary Segmentation Model

7.4.1 The Mumford-Shah Segmentation Model

7.4.2 Asymptotic M.-S. Model I： Sobolev Smoothing

7.4.3 Asymptotic M.-S. Model II： Piecewise Constant

7.4.4 Asymptotic M.-S. Model III： Geodesic Active Contours

7.4.5 Nonuniqueness of M.-S. Segmentation： A 1-D Example

7.4.6 Existence of M.-S. Segmentation

7.4.7 How to Segment Sierpinski Islands

7.4.8 Hidden Symmetries of M.-S. Segmentation

7.4.9 Computational Method I： F-Convergence Approximation

7.4.10 Computational Method II： Level-Set Method

7.5  Multichannel Logical Segmentation

Bibliography

Index