Nonlinear Eigenproblems in Image Processing and Computer Vision (Advances in Computer Vision and Pattern Recognition)

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. The text opens with an introduction to the mathematical background, together with a summary of classical variational algorithms for vision. This is followed by a focus on the foundations and applications of the new multi-scale representation based on non-linear eigenproblems. The book then concludes with a discussion of new numerical techniques for finding nonlinear eigenfunctions, and promising research directions beyond the convex case. Topics and features: Introduces the classical Fourier transform and its associated operator and energy, and asks how these concepts can be generalized in the nonlinear case Reviews the basic mathematical notion, briefly outlining the use of variational and flow-based methods to solve image-processing and computer vision algorithms Describes the properties of the total variation (TV) functional, and how the concept of nonlinear eigenfunctions relate to convex functionals Provides a spectral framework for one-homogeneous functionals, and applies this framework for denoising, texture processing and image fusion Proposes novel ways to solve the nonlinear eigenvalue problem using special flows that converge to eigenfunctions Examines graph-based and nonlocal methods, for which a TV eigenvalue analysis gives rise to strong segmentation, clustering and classification algorithms Presents an approach to generalizing the nonlinear spectral concept beyond the convex case, based on pixel decay analysis Discusses relations to other branches of image processing, such as wavelets and dictionary based methods This original work offers fascinating new insights into established signal processing techniques, integrating deep mathematical concepts from a range of different fields, which will be of great interest to all researchers involved with image processing and computer vision applications, as well as computations for more general scientific problems. Dr. Guy Gilboa is an Assistant Professor in the Electrical Engineering Department at Technion {u2013} Israel Institute of Technology, Haifa, Israel What are Nonlinear Eigenproblems and Why are They Important?......Page 7 Basic Intuition and Examples......Page 9 References......Page 13 Acknowledgements......Page 15 Contents......Page 16 1.1 Reminder of Very Basic Operators and Definitions......Page 20 1.1.1 Integration by Parts (Reminder)......Page 21 1.2 Some Standard Spaces......Page 22 1.3 Euler–Lagrange......Page 23 1.3.1 E–L of Some Functionals......Page 24 1.3.2 Some Useful Examples......Page 25 1.3.4 Norms Without Derivatives......Page 26 1.3.5 Seminorms with Derivatives......Page 27 1.4.1 Convex Function and Functional......Page 28 1.4.3 Subdifferential......Page 29 1.5.1 Definition and Basic Properties......Page 30 References......Page 33 2.1 Variation Modeling by Regularizing Functionals......Page 34 2.1.1 Regularization Energies and Their Respective E-L......Page 36 2.2.1 Gaussian Scale Space......Page 37 2.2.2 Perona–Malik Nonlinear Diffusion......Page 38 2.2.3 Weickert's Anisotropic Diffusion......Page 39 2.2.4 Steady-State Solution......Page 40 2.3.1 Background......Page 41 2.3.2 Early Attempts for Solving the Optical Flow Problem......Page 43 2.3.3 Modern Optical Flow Techniques......Page 44 2.4.2 Mumford–Shah......Page 45 2.4.3 Chan–Vese Model......Page 46 2.4.4 Active Contours......Page 47 2.5.1 Background......Page 48 2.5.2 Graph Laplacian......Page 49 2.5.3 A Nonlocal Mathematical Framework......Page 50 2.5.4 Basic Models......Page 53 References......Page 54 3.1 Strong and Weak Definitions......Page 57 3.3 Definition of BV......Page 58 3.4.2 ROF, TV-L1, and TV Flow......Page 59 References......Page 61 4.1 Introduction......Page 62 4.2 One-Homogeneous Functionals......Page 63 4.3 Properties of Eigenfunction......Page 64 4.4 Eigenfunctions of TV......Page 65 4.4.1 Explicit TV Eigenfunctions in 1D......Page 66 4.5.1 Measure of Affinity of Nonlinear Eigenfunctions......Page 70 References......Page 73 5.2.1 Scale Space Representation......Page 75 5.3 Signal Processing Analogy......Page 77 5.3.2 Spectral Response......Page 79 5.4.1 Variational Representation......Page 82 5.4.2 Scale Space Representation......Page 86 5.4.3 Inverse Scale Space Representation......Page 87 5.4.4 Definitions of the Power Spectrum......Page 89 5.5.1 Basic Conditions on the Regularization......Page 90 5.5.2 Connection Between Spectral Decompositions......Page 93 5.5.3 Orthogonality of the Spectral Components......Page 98 5.5.4 Nonlinear Eigendecompositions......Page 104 References......Page 106 6.2 Simplification and Denoising......Page 108 6.2.1 Denoising with Trained Filters......Page 113 6.3 Multiscale and Spatially Varying Filtering Horesh–Gilboa......Page 115 6.4 Face Fusion and Style Transfer......Page 117 References......Page 119 7.1 Linear Methods......Page 121 7.2 Hein–Buhler......Page 122 7.3 Nossek–Gilboa......Page 123 7.3.1 Flow Main Properties......Page 124 7.3.2 Inverse Flow......Page 127 7.3.4 Properties of the Discrete Flow......Page 128 7.3.5 Normalized Flow......Page 131 7.4 Aujol et al. Method......Page 134 References......Page 136 8.1 Graph Total Variation Analysis......Page 137 8.2 Graph P-Laplacian Operators......Page 138 8.3 The Cheeger Cut......Page 140 8.4 The Graph 1-Laplacian......Page 141 8.5.1 Flow Main Properties......Page 143 8.5.2 Numerical Scheme......Page 144 8.5.3 Algorithm......Page 145 References......Page 146 9.1 General Decomposition Based on Nonlinear Denoisers......Page 147 9.1.2 Inverse Transform, Spectrum, and Filtering......Page 148 9.1.3 Determining the Decay Profiles......Page 149 9.2 Blind Spectral Decomposition......Page 150 9.3 Theoretical Analysis......Page 152 9.3.2 Relation to Known Transforms......Page 153 References......Page 154 10.1 Decomposition into Eigenfunctions......Page 155 10.2.1 Haar Wavelets......Page 156 10.3 Rayleigh Quotients and SVD Decomposition......Page 157 10.4 Sparse Representation by Eigenfunctions......Page 162 10.4.2 Dictionaries from One-Homogeneous Functionals......Page 163 References......Page 164 11.1 Spectral Total Variation Local Time Signatures for Image Manipulation and Fusion......Page 165 11.2 Spectral AATV (Adapted Anisotropic Total Variation) .........Page 167 11.3 TV Spectral Hashing......Page 169 11.4 Some Open Problems......Page 171 Reference......Page 172 A.1 Derivative Operators......Page 173 A.2.2 Evolutions......Page 174 A.3 Basic Numerics for Solving TV......Page 175 A.3.2 Lagged Diffusivity......Page 176 A.4.1 The Proximal Operator......Page 177 A.4.2 Examples of Proximal Functions......Page 178 A.4.5 Chambolle–Pock......Page 179 A.5.1 Basic Discretization......Page 180 A.5.2 Steepest Descent......Page 181 Appendix Glossary......Page 183 References......Page 184 Index......Page 185