Introduction、The Image Society、What Is a Digital Image、About Partial Differential Equations（PDEs）、Detailed Plan、Mathematical Preliminaries、How to Read This Chapter、The Direct Method in the Calculus of Vgriations、Topologies on Banach Spaces、Convexity and Lower Semicontinuity、Rclaxation、Aboutr-Convergence、The Space of Functions of Bounded Variation、Basic Definitions on Measures、Definition ofBV（Ω）、Properties ofBV（Ω）、Convex Functions of Measures、Viscosity Solutions in PDEs等等。
It is surprising when we realize just how much we are surrounded by images．Images allow US not only to perform complex tasks on a daily basis， but also to communicate，transmit information，and represent and under． stand the world around US．Just think、for instance、about digital television， medical imagery，and video surveillance．The tremendous development in information technology accounts for most of this．we are now able to handle more and more data．Many day．to-day tasks are now fully or partially accomplished with the help of computers．Whenever images are involved we are entering the domains of computer vision and image processing．
The requirements for this are reliability and speed．Efficient algorithms have to be proposed to process these digital data．It is also important to rely on a well-established theory to iustifv the well-founded nature of the methodology．
Among the numerous approaches that have been suggested，we focus on partial difierential equations(PDEs)，and variational approaches in this book．Traditionally applied in physics．these methods have been successfully and widely transferred to computer vision over the last decade．One of the main interests in using PDEs iS that the theory behind the concept iS well established．Of course．PDEs are written in a continuous setting referring to analogue images， and once the existence and the uniqueness have been proven．we need to discretize them in order to find a numerical solution．It iS our conviction that reasoning within a continuous frame work makes the underStanding of physical realities easier and stimulates the intuition necessary to propose new models．We hope that this book will illustrate this idea effectively．
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 Image7
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 Vgriations
2.1.1 Topologies on Banach Spaces
2.1.2 Convexity and Lower Semicontinuity
2.1.3 Rclaxat.ion
2.1.4 About r-Convergence
2.2 The Space of Functions of Bounded Variation
2.2.1 Basic Definitions on Measures
2.2.2 Definition ofBV（Ω）
2.2.3 Properties ofBV（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 Differential 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 0ther 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了heorem
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
3.2.6 Some Remarks on the Nonconvex CaSe
3.3 PDE-BaSed Methods
3.3.1 Smoothing PDEs
The Heat Equation
Nonlinear DiRusion
The Alvarez-Guichard-Lions-Morel
Scale Space Theory
Weickerts Approach
Surface Based Approaches
3.3.2 Smoothing-Enhancing PDEs
The Perona and Malik Model
Regutarization 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 ofCharacteristics
Comments on the Shock-Filter Equation
3.3.4 NeighborbOOd 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 a1.Approach
The Chan and Shen Curvature-Driven Diffusion 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 C：ontribution
The Aujol et a1.Approach
Study of the Asymptotic Case
Back to Meyers 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 Vriational 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 FXtended Nbtion of Grudieut
A Introduction to Finite Digerence Methods
B Experiment Yourself！
References
Index
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