Variational Methods : In Imaging and Geometric Control.
Bergounioux, Maïtine.
Variational Methods : In Imaging and Geometric Control. - 1st ed. - 1 online resource (538 pages) - Radon Series on Computational and Applied Mathematics Series ; v.18 . - Radon Series on Computational and Applied Mathematics Series .
Intro -- Contents -- Part I -- 1 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 -- 2 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 -- 3 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 G -- 3.4.1 Basis on T?id (G) -- 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 -- 4 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 -- 5 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 -- 6 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 -- 7 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 -- 7.6 Conclusion and outlook -- 7.6.1 Mass preservation -- 7.6.2 Higher dimensions -- 7.6.3 Joint models -- 7.6.4 Large displacements -- 8 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.
9783110430394
Image processing--Mathematical models.
Electronic books.
TA1637.5.V37 2017
Variational Methods : In Imaging and Geometric Control. - 1st ed. - 1 online resource (538 pages) - Radon Series on Computational and Applied Mathematics Series ; v.18 . - Radon Series on Computational and Applied Mathematics Series .
Intro -- Contents -- Part I -- 1 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 -- 2 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 -- 3 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 G -- 3.4.1 Basis on T?id (G) -- 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 -- 4 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 -- 5 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 -- 6 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 -- 7 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 -- 7.6 Conclusion and outlook -- 7.6.1 Mass preservation -- 7.6.2 Higher dimensions -- 7.6.3 Joint models -- 7.6.4 Large displacements -- 8 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.
9783110430394
Image processing--Mathematical models.
Electronic books.
TA1637.5.V37 2017