Nonlinear Eigenproblems in Image Processing and Computer Vision
This unique text/reference presents a fresh look at nonlinear processing through nonlinear eigenvalue analysis, highlighting how one-homogeneous convex functionals can induce nonlinear operators that can be analyzed within an eigenvalue framework.
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| Main Author | |
|---|---|
| Format | eBook |
| Language | English |
| Published |
Cham
Springer Nature
2018
Springer International Publishing AG Springer International Publishing |
| Edition | 1 |
| Series | Advances in Computer Vision and Pattern Recognition |
| Subjects | |
| Online Access | Get full text |
| ISBN | 9783319758473 3319758470 9783319758466 3319758462 |
| ISSN | 2191-6586 2191-6594 |
| DOI | 10.1007/978-3-319-75847-3 |
Cover
Table of Contents:
- 3.4.2 ROF, TV-L1, and TV Flow -- References -- 4 Eigenfunctions of One-Homogeneous Functionals -- 4.1 Introduction -- 4.2 One-Homogeneous Functionals -- 4.3 Properties of Eigenfunction -- 4.4 Eigenfunctions of TV -- 4.4.1 Explicit TV Eigenfunctions in 1D -- 4.5 Pseudo-Eigenfunctions -- 4.5.1 Measure of Affinity of Nonlinear Eigenfunctions -- References -- 5 Spectral One-Homogeneous Framework -- 5.1 Preliminary Definitions and Settings -- 5.2 Spectral Representations -- 5.2.1 Scale Space Representation -- 5.3 Signal Processing Analogy -- 5.3.1 Nonlinear Ideal Filters -- 5.3.2 Spectral Response -- 5.4 Theoretical Analysis and Properties -- 5.4.1 Variational Representation -- 5.4.2 Scale Space Representation -- 5.4.3 Inverse Scale Space Representation -- 5.4.4 Definitions of the Power Spectrum -- 5.5 Analysis of the Spectral Decompositions -- 5.5.1 Basic Conditions on the Regularization -- 5.5.2 Connection Between Spectral Decompositions -- 5.5.3 Orthogonality of the Spectral Components -- 5.5.4 Nonlinear Eigendecompositions -- References -- 6 Applications Using Nonlinear Spectral Processing -- 6.1 Generalized Filters -- 6.1.1 Basic Image Manipulation -- 6.2 Simplification and Denoising -- 6.2.1 Denoising with Trained Filters -- 6.3 Multiscale and Spatially Varying Filtering Horesh-Gilboa -- 6.4 Face Fusion and Style Transfer -- References -- 7 Numerical Methods for Finding Eigenfunctions -- 7.1 Linear Methods -- 7.2 Hein-Buhler -- 7.3 Nossek-Gilboa -- 7.3.1 Flow Main Properties -- 7.3.2 Inverse Flow -- 7.3.3 Discrete Time Flow -- 7.3.4 Properties of the Discrete Flow -- 7.3.5 Normalized Flow -- 7.4 Aujol et al. Method -- References -- 8 Graph and Nonlocal Framework -- 8.1 Graph Total Variation Analysis -- 8.2 Graph P-Laplacian Operators -- 8.3 The Cheeger Cut -- 8.4 The Graph 1-Laplacian -- 8.5 The p-flow -- 8.5.1 Flow Main Properties
- Intro -- Preface -- What are Nonlinear Eigenproblems and Why are They Important? -- Basic Intuition and Examples -- What is Covered in This Book? -- References -- Acknowledgements -- Contents -- 1 Mathematical Preliminaries -- 1.1 Reminder of Very Basic Operators and Definitions -- 1.1.1 Integration by Parts (Reminder) -- 1.1.2 Distributions (Reminder) -- 1.2 Some Standard Spaces -- 1.3 Euler-Lagrange -- 1.3.1 E-L of Some Functionals -- 1.3.2 Some Useful Examples -- 1.3.3 E-L of Common Fidelity Terms -- 1.3.4 Norms Without Derivatives -- 1.3.5 Seminorms with Derivatives -- 1.4 Convex Functionals -- 1.4.1 Convex Function and Functional -- 1.4.2 Why Convex Functions Are Good? -- 1.4.3 Subdifferential -- 1.4.4 Duality-Legendre-Fenchel Transform -- 1.5 One-Homogeneous Functionals -- 1.5.1 Definition and Basic Properties -- References -- 2 Variational Methods in Image Processing -- 2.1 Variation Modeling by Regularizing Functionals -- 2.1.1 Regularization Energies and Their Respective E-L -- 2.2 Nonlinear PDEs -- 2.2.1 Gaussian Scale Space -- 2.2.2 Perona-Malik Nonlinear Diffusion -- 2.2.3 Weickert's Anisotropic Diffusion -- 2.2.4 Steady-State Solution -- 2.2.5 Inverse Scale Space -- 2.3 Optical Flow and Registration -- 2.3.1 Background -- 2.3.2 Early Attempts for Solving the Optical Flow Problem -- 2.3.3 Modern Optical Flow Techniques -- 2.4 Segmentation and Clustering -- 2.4.1 The Goal of Segmentation -- 2.4.2 Mumford-Shah -- 2.4.3 Chan-Vese Model -- 2.4.4 Active Contours -- 2.5 Patch-Based and Nonlocal Models -- 2.5.1 Background -- 2.5.2 Graph Laplacian -- 2.5.3 A Nonlocal Mathematical Framework -- 2.5.4 Basic Models -- References -- 3 Total Variation and Its Properties -- 3.1 Strong and Weak Definitions -- 3.2 Co-area Formula -- 3.3 Definition of BV -- 3.4 Basic Concepts Related to TV -- 3.4.1 Isotropic and Anisotropic TV
- 8.5.2 Numerical Scheme -- 8.5.3 Algorithm -- References -- 9 Beyond Convex Analysis-Decompositions with Nonlinear Flows -- 9.1 General Decomposition Based on Nonlinear Denoisers -- 9.1.1 A Spectral Transform -- 9.1.2 Inverse Transform, Spectrum, and Filtering -- 9.1.3 Determining the Decay Profiles -- 9.2 Blind Spectral Decomposition -- 9.3 Theoretical Analysis -- 9.3.1 Generalized Eigenvectors -- 9.3.2 Relation to Known Transforms -- References -- 10 Relations to Other Decomposition Methods -- 10.1 Decomposition into Eigenfunctions -- 10.2 Wavelets and Hard Thresholding -- 10.2.1 Haar Wavelets -- 10.3 Rayleigh Quotients and SVD Decomposition -- 10.4 Sparse Representation by Eigenfunctions -- 10.4.1 Total Variation Dictionaries -- 10.4.2 Dictionaries from One-Homogeneous Functionals -- References -- 11 Future Directions -- 11.1 Spectral Total Variation Local Time Signatures for Image Manipulation and Fusion -- 11.2 Spectral AATV (Adapted Anisotropic Total Variation) … -- 11.3 TV Spectral Hashing -- 11.4 Some Open Problems -- Reference -- A Numerical Schemes -- A.1 Derivative Operators -- A.2 Discretization of PDE's -- A.2.1 Discretized Differential Operators -- A.2.2 Evolutions -- A.2.3 CFL Condition -- A.3 Basic Numerics for Solving TV -- A.3.1 Explicit Method -- A.3.2 Lagged Diffusivity -- A.3.3 Chambolle's Projection Algorithm -- A.4 Modern Optimization Methods -- A.4.1 The Proximal Operator -- A.4.2 Examples of Proximal Functions -- A.4.3 ADMM -- A.4.4 FISTA -- A.4.5 Chambolle-Pock -- A.5 Nonlocal Models -- A.5.1 Basic Discretization -- A.5.2 Steepest Descent -- Appendix Glossary -- References -- Index