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Nonlinear Eigenproblems in image processing and computer vision

Author: Guy Gilboa
Publisher: Cham, Switzerland : Springer, 2018.
Series: Advances in computer vision and pattern recognition
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Gilboa, Guy.
Nonlinear Eigenproblems in image processing and computer vision.
Cham, Switzerland : Springer, 2018
(OCoLC)1020028206
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Guy Gilboa
ISBN: 9783319758473 3319758470
OCLC Number: 1030438586
Description: 1 online resource (xx, 172 pages) : illustrations (some color)
Contents: 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 Functional1.4.2 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 Background2.3.2 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. 3.4.2 ROF, TV-L1, and TV FlowReferences; 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 Properties5.4.1 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.
Series Title: Advances in computer vision and pattern recognition
Responsibility: Guy Gilboa.

Abstract:

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  Read more...

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