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by Thomas Haberkorn, Jean-Baptiste Caillau, Christoph Schnörr, Gabriel Peyré, Maïti
Variational Methods
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Title Page
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Contents
Part I
Second-order decomposition model for image processing: numerical experimentation
1.1 Introduction
1.2 Presentation of the model
1.3 Numerical aspects
1.3.1 Discretized problem and algorithm
1.3.2 Examples
1.3.3 Initialization process
1.3.4 Convergence
1.3.5 Sensitivity with respect to sampling and quantification
1.3.6 Sensitivity with respect to parameters
1.4 Conclusion
Optimizing spatial and tonal data for PDE-based inpainting
2.1 Introduction
2.2 A review of PDE-based image compression
2.2.1 Data optimization
2.2.2 Finding good inpainting operators
2.2.3 Storing the data
2.2.4 Feature-based methods
2.2.5 Fast algorithms and real-time aspects
2.2.6 Hybrid image compression methods
2.2.7 Modifications, extensions and applications
2.2.8 Relations to other methods
2.3 Inpainting with homogeneous diffusion
2.4 Optimization strategies in 1D
2.4.1 Optimal knots for interpolating convex functions
2.4.2 Optimal knots for approximating convex functions
2.5 Optimization strategies in 2D
2.5.1 Optimizing spatial data
2.5.2 Optimizing tonal data
2.6 Extensions to other inpainting operators
2.6.1 Optimizing spatial data
2.6.2 Optimizing tonal data
2.7 Summary and conclusions
Image registration using phase–amplitude separation
3.1 Introduction
3.1.1 Current literature
3.1.2 Our approach
3.2 Definition of phase–amplitude components
3.2.1 q-Map and amplitude distance
3.2.2 Relative phase and image registration
3.3 Properties of registration framework
3.4 Gradient method for optimization over Γ
3.4.1 Basis on
3.4.2 Mean image and group-wise registration
3.5 Experiments
3.5.1 Pairwise image registration
3.5.2 Registering multiple images
3.5.3 Image classification
3.6 Conclusion
Rotation invariance in exemplar-based image inpainting
4.1 Introduction to inpainting
4.1.1 The inpainting problem
4.1.2 Aims of this work
4.1.3 Notation
4.2 Rotation invariant image pattern recognition
4.2.1 Patch error functions
4.2.2 Circular harmonics basis
4.2.3 Mutual angle detection algorithms
4.2.4 Rotation invariant L2-error using the circular harmonics basis
4.2.5 Rotation invariant gradient-based L2-errors and the CH-basis
4.3 Rotation invariant exemplar-based inpainting
4.3.1 Patch non-local means
4.3.2 Patch non-local Poisson
4.3.3 Numerical experiments
4.4 Discussion and analysis
4.4.1 Proof of convergence
4.4.2 Analysis of E∇,T
4.4.3 Conclusion and future perspectives
Convective regularization for optical flow
5.1 Introduction
5.2 Model
5.2.1 Convective acceleration
5.2.2 Convective regularization
5.2.3 Data term and contrast invariance
5.3 Numerical solution
5.4 Experiments
5.5 Conclusion
A variational method for quantitative photoacoustic tomography with piecewise constant coefficients
6.1 Quantitative photoacoustic tomography
6.1.1 Introduction
6.1.2 Contributions of this article
6.2 Recovery of piecewise constant coefficients
6.3 A Mumford–Shah-like functional for qPAT
6.3.1 Existence of minimizers
6.3.2 Approximation
6.3.3 Minimization
6.4 Implementation and numerical results
A Special functions of bounded variation and the SBV-compactness theorem
On optical flow models for variational motion estimation
7.1 Introduction
7.2 Models
7.2.1 Variational models with gradient regularization
7.2.2 Extension of the regularizer
7.2.3 Bregman iterations
7.3 Analysis
7.3.1 Existence of minimizers
7.3.2 Quantitative estimates
7.4 Numerical solution
7.4.1 Primal–dual algorithm
7.4.2 Discretization and parameters
7.5 Results
7.5.1 Error measures for velocity fields
7.5.2 Evaluation
7.6 Conclusion and outlook
7.6.1 Mass preservation
7.6.2 Higher dimensions
7.6.3 Joint models
7.6.4 Large displacements
Bilevel approaches for learning of variational imaging models
8.1 Overview of learning in variational imaging
8.2 The learning model and its analysis in function space
8.2.1 The abstract model
8.2.2 Existence and structure: L2-squared cost and fidelity
8.2.3 Optimality conditions
8.3 Numerical optimization of the learning problem
8.3.1 Adjoint-based methods
8.3.2 Dynamic sampling
8.4 Learning the image model
8.4.1 Total variation-type regularization
8.4.2 Optimal parameter choice for TV-type regularization
8.5 Learning the data model
8.5.1 Variational noise models
8.5.2 Single noise estimation
8.5.3 Multiple noise estimation
8.6 Conclusion and outlook
Part II
Non-degenerate forms of the generalized Euler–Lagrange condition for state-constrained optimal control problems
9.1 Introduction
9.2 Main result
9.3 Proof of Theorem 2.1
9.4 Proof of Lemma 3.2
9.5 Example
The Purcell three-link swimmer: some geometric and numerical aspects related to periodic optimal controls
10.1 Introduction
10.2 First- and second-order optimality conditions
10.3 The Purcell three-link swimmer
10.3.1 Mathematical model
10.4 Local analysis for the three-link Purcell swimmer
10.4.1 Computations of the nilpotent approximation
10.4.2 Integration of extremal trajectories
10.5 Numerical results
10.5.1 Nilpotent approximation
10.5.2 True mechanical system
10.5.3 The Purcell swimmer in a round swimming pool
10.6 Conclusions and future work
Controllability of Keplerian motion with low-thrust control systems
11.1 Introduction
11.2 Notations and definitions
11.2.1 Dynamics
11.2.2 Study of the drift vector field in ?
11.2.3 Admissible controlled trajectory of Σsat
11.2.4 Controlled problems in ?
11.3 Controllability
11.3.1 Controllability for OTP
11.3.2 Controllability for OIP
11.3.3 Controllability for DOP
11.4 Numerical examples
11.4.1 A numerical example for OIP
11.4.2 A numerical example for DOP
11.5 Conclusion
11.6 Appendix
Higher variational equation techniques for the integrability of homogeneous potentials
12.1 Introduction: integrable systems
12.2 An algebraic point of view
12.2.1 Algebraic presentation of a Hamiltonian system
12.2.2 First-order variational equations
12.2.3 Differential Galois theory
12.3 Introduction to Morales–Ramis theorem
12.3.1 The Morales–Ramis theorem
12.3.2 Homogeneous potentials
12.3.3 Higher variational equations
12.4 Application to parametrized potentials
12.4.1 Space of germs of integrable potentials
12.4.2 Eigenvalue bounding of some n-body problems
Introduction to KAM theory with a view to celestial mechanics
13.1 Twisted conjugacy normal form
13.2 One step of the Newton algorithm
13.3 Inverse function theorem
13.4 Local uniqueness and regularity of the normal form
13.5 Conditional conjugacy
13.6 Invariant torus with prescribed frequency
13.7 Invariant tori with unprescribed frequencies
13.8 Symmetries
13.9 Lower dimensional tori
13.10 Example in the spatial three-body problem
A Isotropy of invariant tori
B Two basic estimates
C Interpolation of spaces of analytic functions
Invariants of contact sub-pseudo-Riemannian structures and Einstein–Weyl geometry
14.1 Introduction
14.2 Dimension 3
14.3 Einstein–Weyl geometry
14.4 Dimension 2n + 1
14.5 Contact sub-pseudo-Riemannian symmetries
14.6 Appendix: Isometries in dimension 5
Time-optimal control for a perturbed Brockett integrator
15.1 Introduction
15.2 Controllability and time-optimal controllability
15.3 An approximate linearized time-optimal control problem
15.4 Numerical computation of a time-optimal trajectory
15.4.1 Finite-dimensional minimization problem
15.4.2 Numerical example
15.5 Application to time-optimal micro-swimmers
15.5.1 Modeling and problem formulation
15.5.2 Numerical computation of a time-optimal trajectory
15.6 Conclusion
Twist maps and Arnold diffusion for diffeomorphisms
16.1 From Arnold diffusion to twist maps
16.2 Setting and main result
16.3 Proof of Theorem 1
16.3.1 Choice of ε > 0
16.3.2 Implementation of Moeckel’s method
16.3.3 Normally hyperbolic shadowing
16.3.4 Conclusion of the proof of Theorem 1
A Hamiltonian approach to sufficiency in optimal control with minimal regularity conditions: Part I
17.1 Introduction
17.1.1 The problem
17.2 Notations and preliminary results
17.2.1 Symplectic notations
17.2.2 The Pontryagin maximum principle
17.3 A Hamiltonian approach to optimality
17.3.1 The Cartan form
17.3.2 The super-Hamiltonian and its properties
17.3.3 Abstract sufficient optimality conditions
17.3.4 The minimum time problem
17.4 Final comments
Index
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