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Table of contents
1 Introduction
1.1 Viscosity Solutions Theory for Integro-Differential Equations
1.1.1 Lévy Processes and Integro-Differential Equations (Chapter 2)
1.1.2 StrongMaximum Principle (Chapter 3)
1.1.3 Regularity of Solutions (Chapter 4)
1.1.4 Work in Progress and Future Plans
1.2 Image Restoration and Visualization by CurvatureMotions
1.2.1 Algorithms for CurvatureMotions (Chapter 5)
1.2.2 From Curve Shortening to Image CurvatureMotion (Chapter 6)
1.2.3 Level Lines Shortening Algorithm (Chapter 7)
1.2.4 Work in Progress and Future Plans
1.3 Publications related to the thesis
2 Levy Processes. Integro-Differential Equations
2.1 Introduction
2.2 Lévy processes
2.2.1 The Structure of Lévy processes. Lévy-Khintchine Formula
2.2.2 Examples of Lévy Processes
2.2.3 The Lévy-Itô Decomposition
2.2.4 Integro-Differential Operators
2.3 Integro-Differential Equations
2.3.1 Existence and Comparison Results
2.3.2 Regularity Theory for Fully Nonlinear Integro-Differential Equations
2.4 Discussion
3 StrongMaximum Principle
3.1 Introduction
3.2 StrongMaximum Principle – General Nonlocal Operators
3.2.1 Horizontal Propagation – Translations ofMeasure Supports
3.2.2 Horizontal Propagation – Nondegeneracy Conditions
3.2.3 Local Vertical Propagation
3.2.4 StrongMaximum Principle
3.3 StrongMaximum Principle for Lévy-Itô operators
3.4 Examples
3.4.1 Horizontal Propagation – Translations ofMeasure Supports
3.4.2 StrongMaximum Principle – Nondegenerate Nonlocal
3.4.3 StrongMaximum Principle coming from Local Diffusion Terms
3.4.4 Strong Maximum Principle for Mixed Differential-Nonlocal terms
3.5 Strong Comparison Principl
3.6 Appendix
3.7 Conclusion
4 Lipschitz RegularityMixed PIDES
4.1 Introduction
4.2 Notations and Assumptions
4.2.1 Viscosity Solutions for Integro-Differential Equations
4.2.2 Ellipticity Growth Conditions
4.2.3 LévyMeasures for General Nonlocal Operators
4.2.4 LévyMeasures for Lévy-Itô Operators
4.3 Lipschitz Continuity of Viscosity Solutions
4.3.1 Partial Regularity Results
4.3.2 Global Regularity
4.4 Examples and Discussion on Assumptions
4.4.1 Classical Nonlinearities
4.4.2 A Toy-Model for the mixed case
4.4.3 Mixed Integro-Differential Equations with First-Order Terms
4.5 Some Extensions of the Regularity Results
4.5.1 Non-periodic Setting
4.5.2 Parabolic Integro-Differential Equations
4.5.3 Bellman-Isaacs Equations
4.5.4 Multiple Nonlinearities
4.6 Estimates for Integro – Differential Operators
4.6.1 General Nonlocal Operators
4.6.2 Lévy-Itô Operators
4.7 Appendix
4.8 Conclusion
5 Curves and Curvatures
5.1 Introduction
5.2 Curvature Scale Spaces
5.2.1 Curvatures
5.2.2 Curve Evolutions
5.2.3 Image Evolutions
5.2.4 Connection
5.3 Curvature algorithms
5.3.1 Algorithms on Curves
5.3.2 Algorithms on Sets
5.3.3 Algorithms on Images
5.4 Discussion
6 Equivalence LLS -MCM
6.1 Introduction
6.2 Modeling Level Lines Shortening
6.2.1 Crowns of Jordan Curves
6.2.2 The Class of Very Simple Functions
6.3 Level Lines Shortening Operator
6.3.1 LLS Semigroup Operator
6.3.2 Properties of Level Lines Shortening
6.4 Equivalence with the CurvatureMotions
6.4.1 Mean CurvatureMotion
6.4.2 Affine CurvatureMotion
6.5 Conclusion
7 LLS algorithm and Curvature Estimates
7.1 Introduction
7.2 The Level Lines Shortening Algorithm
7.2.1 Bilinear Interpolation
7.2.2 Direct Extraction Algorithm of Bilinear Level Lines
7.2.3 Independent Evolution of All Level Lines
7.2.4 Image Reconstruction from a Tree of Level Lines
7.3 Numerical Properties of the LLS Numerical Chain
7.3.1 Fixed Point Property
7.3.2 Local Comparison Principle and Regularity
7.3.3 JPEG Artifacts Reduction on Color Images
7.3.4 AccurateMean Curvature Evolution
7.4 An Image CurvatureMicroscope
7.4.1 Discrete curvature for a polygonal line
7.4.2 Curvature map
7.4.3 Comparison with FDSes
7.4.4 CurvatureMicroscope
7.5 Image Restoration and Vizualization
7.5.1 Founding Example: Attneave’s cat
7.5.2 Geometric Shapes
7.5.3 Graphics and Aliasing
7.5.4 Pre-attentively Undiscriminable Textons
7.5.5 BacteriaMorphologies
7.5.6 Topography
7.5.7 Textures
7.5.8 Paintings
7.5.9 Text Processing
7.5.10 Fingerprints Restoration and Discrimination
7.6 Conclusion
