Foreword Preface to the Second Edition Preface to the First Edition Guide to the Main Mathematical Concepts and Their Application Notation and Symbols 1 Introduction 1.1 The Image Society 1.2 What Is a Digital Image 1.3 About Partial Differential Equations (PDEs) 1.4 Detailed Plan 2 Mathematical Preliminaries How to Read This Chapter 2.1 The Direct Method in the Calculus of Variations 2.1.1 Topologies on Banach Spaces 2.1.2 Convexity and Lower Semicontinuity 2.1.3 Relaxation 2.1.4 About T-Convergence 2.2 The Space of Functions of Bounded Variation 2.2.1 Basic Definitions on Measures 2.2.2 Definition of BV (Ω) 2.2.3 Properties of BV (Ω) 2.2.4 Convex Functions of Measures 2.3 Viscosity Solutions in PDEs 2.3.1 About the Eikonal Equation 2.3.2 Definition of Viscosity Solutions 2.3.3 About the Existence 2.3.4 About the Uniqueness 2.4 Elements of Diferential Geometry: Curvature 2.4.1 Parametrized Curves 2.4.2 Curves as Isolevel of a Function u 2.4.3 Images as Surfaces 2.5 Other Classical Results Used in This Book 2.5.1 Inequalities 2.5.2 Calculus Facts 2.5.3 About Convolution and Smoothing 2.5.4 Uniform Convergence 2.5.5 Dominated Convergence Theorem 2.5.6 Well-Posed Problems 3 Image Restoration How to Read This Chapter 3.1 Image Degradation 3.2 The Energy Method 3.2.1 An Inverse Problem 3.2.2 Regularization of the Problem 3.2.3 Existence and Uniqueness of a Solution for the Minimization Problem 3.2.4 Toward the Numerical Approximation The Projection Approach The Half-Quadratic Minimization Approach 3.2.5 Some Invariances and the Role of λ 3.2.6 Some Remarks on the Nonconvex Case
3.3 PDE-Based Methods 3.3.1 Smoothing PDEs The Heat Equation Nonlinear Diffusion The Alvarez-Guichard-Lions-Morel Scale Space Theory Weickert's Approach Surface Based Approaches 3.3.2 Smoothing-Enhancing PDEs The Perona and Malik Model Regularization of the Perona and Malik Model: Catte et al 3.3.3 Enhancing PDEs The Osher and Rudin Shock Filters A Case Study: Construction of a Solution by the Method of Characteristics Comments on the Shock-Filter Equation 3.3.4 Neighborhood Filters, Nonlocal Means Algorithm, and PDEs Neighborhood Filters How to Suppress the Staircase Effect Nonlocal Means Filter (NL-Means) 4 The Segmentation Problem How to Read This Chapter 4.1 Definition and Objectives 4.2 The Mumford and Shah Functional 4.2.1 A Minimization Problem 4.2.2 The Mathematical Framework for the Existence of a Solution 4.2.3 Regularity of the Edge Set 4.2.4 Approximations of the Mumford and Shah Functional 4.2.5 Experimental Results 4.3 Geodesic Active Contours and the Level-Set Method 4.3.1 The Kass-Witkin-Terzopoulos model 4.3.2 The Geodesic Active Contours Model 4.3.3 The Level-Set Method 4.3.4 The Reinitialization Equation Characterization of the Distance Function Existence and Uniqueness 4.3.5 Experimental Results 4.3.6 About Some Recent Advances Global Stopping Criterion Toward More General Shape Representation 5 Other Challenging Applications How to Read This Chapter 5.1 Reinventing Some Image Parts by Inpainting 5.1.1 Introduction 5.1.2 Variational Models The Masnou and Morel Approach The Ballester et al.Approach The Chan and Shen Total Variation Minimization Approach 5.1.3 PDE-Based Approaches The Bertalmio et al.Approach The Chan and Shen Curvature-Driven Difusion Approach
5.1.4 Discussion 5.2 Decomposing an Image into Geometry and Texture 5.2.1 Introduction 5.2.2 A Space for Modeling Oscillating Patterns 5.2.3 Meyer's Model 5.2.4 An Algorithm to Solve Meyer's Model Prior Numerical Contribution The Aujol et al.Approach Study of the Asymptotic Case Back to Meyer's Model 5.2.5 Experimental Results Denoising Capabilities Dealing With Texture 5.2.6 About Some Recent Advances 5.3 Sequence Analysis 5.3.1 Introduction 5.3.2 The Optical Flow: An Apparent Motion The Optical Flow Constraint (OFC) Solving the Aperture Problem Overview of a Discontinuity-Preserving Variational Approach Alternatives to the OFC 5.3.3 Sequence Segmentation Introduction A Variational Formulation Mathematical Study of the Time-Sampled Energy Experiments 5.3.4 Sequence Restoration Principles of Video Inpainting Total Variation (TV) Minimization Approach Motion Compensated (MC) Inpainting 5.4 Image Classification 5.4.1 Introduction 5.4.2 A Level-Set Approach for Image Classification 5.4.3 A Variational Model for Image Classification and Restoration 5.5 Vector-Valued Images 5.5.1 Introduction 5.5.2 An Extended Notion of Gradient 5.5.3 The Energy Method 5.5.4 PDE-Based Methods A Introduction to Finite Difference Methods How to Read This Chapter A.1 Definitions and Theoretical Considerations Illustrated by the 1-D Parabolic Heat Equation A.1.1 Getting Started A.1.2 Convergence A.1.3 The Lax Theorem A.1.4 Consistency A.1.5 Stability A.2 Hyperbolic Equations A.3 Diference Schemes in Image Analysis A.3.1 Getting Started
A.3.2 Image Restoration by Energy Minimization A.3.3 Image Enhancement by the Osher and Rudin Shock Filters A.3.4 Curve Evolution with the Level-Set Method Mean Curvature Motion Constant Speed Evolution The Pure Advection Equation Image Segmentation by the Geodesic Active Contour Model B Experiment Yourself How to Read This Chapter B.1 The CImg Library B.2 What Is Available Online References Index
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