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BL
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| Record Number
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864985
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| Main Entry
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Gilboa, Guy
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| Title & Author
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Nonlinear Eigenproblems in image processing and computer vision /\ Guy Gilboa.
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| Publication Statement
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Cham, Switzerland :: Springer,, 2018.
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| Series Statement
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Advances in computer vision and pattern recognition,
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| Page. NO
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1 online resource (xx, 172 pages) :: illustrations (some color)
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| ISBN
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3319758470
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: 9783319758473
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3319758462
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9783319758466
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| Bibliographies/Indexes
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Includes bibliographical references.
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| Contents
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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.
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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.
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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.
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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.
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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.
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| Abstract
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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 - Israel Institute of Technology, Haifa, Israel.
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| Subject
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Computer vision-- Mathematics.
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Eigenfunctions.
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Image processing-- Digital techniques-- Mathematics.
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Calculus of variations.
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Eigenfunctions.
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Image processing-- Digital techniques-- Mathematics.
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Image processing.
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Imaging systems technology.
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Mathematical modelling.
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MATHEMATICS-- Calculus.
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MATHEMATICS-- Mathematical Analysis.
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| Subject
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Maths for computer scientists.
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| Dewey Classification
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515/.43
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| LC Classification
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QA371
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