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
