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Shearlets : Multiscale Analysis for Multivariate Data / edited by Gitta Kutyniok, Demetrio Labate
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Applied and Numerical Harmonic Analysis | SpringerLink BücherPublisher: Boston, Mass. [u.a.] : Birkhäuser / Springer, 2012Description: Online-Ressource (XIX, 328p. 50 illus., 19 illus. in color, digital)ISBN:- 9780817683160
- 515.2433 515/.2433
- 515.2433
- QA403.5-404.5
Contents:
Summary: Introduction -- Shearlets and Microlocal Analysis -- Analysis and Identification of Multidimensional Singularities using the Continuous Shearlet Transform -- Multivariate Shearlet Transform, Shearlet Coorbit Spaces and their Structural Properties -- Shearlets and Optimally Sparse Approximations -- Shearlet Multiresolution and Multiple Refinement -- Digital Shearlet Transforms -- Imaging Applications. .Summary: Over the last 20 years, multiscale methods and wavelets have revolutionized the field of applied mathematics by providing efficient means for encoding isotropic phenomena. Directional multiscale systems, particularly shearlets, are currently having the same dramatic impact on the encoding of multivariate signals, which are usually dominated by anisotropic features. Since its introduction about six years ago, the theory of shearlets has rapidly developed and gained wide recognition as the superior approach for multiscale analysis of multivariate signals, providing a truly unified treatment of both the continuum and the digital setting. By now, this research field has reached full maturity, with deep mathematical results, efficient numerical methods, and a variety of high-impact applications. Edited by the topic's two main pioneers, this volume systematically surveys the theory and applications of shearlets. Following a general survey of the subject, carefully selected contributions explore the current state of the field in greater detail. Specific areas covered include: * analysis of anisotropic features; * sparse approximations of multivariate data; * shearlet smoothness spaces; * numerical implementations; * applications to image processing. Shearlets is aimed at graduate students and researchers in the areas of applied mathematics, computer science, engineering, and any other field dealing with the development and applications of highly efficient methodologies for processing multivariate data. As the first monograph in the literature to survey shearlets, this volume offers both a unique state-of-the-art resource for scientists dealing with advanced multiscale methods and a supplemental textbook for graduate courses in applied harmonic analysis.PPN: PPN: 1651396604Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Shearlets; ANHA Series Preface; Preface; Contents; Contributors; Introduction to Shearlets; 1 Introduction; 2 The Rise of Shearlets; 2.1 The Role of Applied Harmonic Analysis; 2.2 Wavelets and Beyond; 3 Notation and Background Material; 3.1 Fourier Analysis; 3.2 Modeling of Signal Classes; 3.3 Frame Theory; 3.4 Wavelets; 3.5 Wavelets for Multivariate Data and Their Limitations; 4 Continuous Shearlet Systems; 4.1 Continuous Shearlet Systems and the Shearlet Group; 4.2 The Continuous Shearlet Transform; 4.3 Cone-Adapted Continuous Shearlet Systems
4.4 The Cone-Adapted Continuous Shearlet Transform4.5 Microlocal Properties and Characterization of Singularities; 5 Discrete Shearlet Systems; 5.1 Discrete Shearlet Systems and Transforms; 5.2 Cone-Adapted Discrete Shearlet Systems and Transforms; 5.3 Compactly Supported Shearlets; 5.4 Sparse Approximations by Shearlets; 5.5 Shearlet Function Spaces; 5.6 Extensions and Generalizations; 6 Algorithmic Implementations of the Shearlet Transform; 6.1 Fourier-Based Implementations; 6.2 Spatial-Domain-Based Implementations; 7 Shearlets in Applications; References; Shearlets and Microlocal Analysis
1 Introduction1.1 Notation; 1.2 Getting to Know the Wavefront Set; 1.2.1 Shearlets and the wavefront set; 1.2.2 Wavelets and the wavefront set; 1.3 Contributions; 1.4 Other Ways to Characterize the Wavefront Set; 2 Reproduction Formulas; 3 Resolution of the Wavefront Set; 3.1 A Direct Theorem; 3.2 Properties of the Wavefront Set; 3.3 Proof of the Main Result; References; Analysis and Identification of Multidimensional SingularitiesUsing the Continuous Shearlet Transform; 1 Introduction; 1.1 Example: Line Singularity; 1.2 General Singularities; 2 Analysis of Step Singularities (2D)
2.1 Shearlet Analysis of Circular Edges2.2 General 2D Boundaries; 2.3 Proofs of Theorems 2 and 3; 2.4 Extensions and Generalizations; 3 Extension to Higher Dimensions; 3.1 3D Continuous Shearlet Transform; 3.2 Characterization of 3D Boundaries; References; Multivariate Shearlet Transform, Shearlet Coorbit Spaces and Their Structural Properties; 1 Introduction; 2 Multivariate Continuous Shearlet Transform; 2.1 Unitary Representations of the Shearlet Group; 2.2 Square Integrable Representations of the Shearlet Group; 2.3 Continuous Shearlet Transform; 3 General Concept of Coorbit Space Theory
3.1 General Coorbit Spaces3.2 Atomic Decompositions and Banach Frames; 4 Multivariate Shearlet Coorbit Theory; 4.1 Shearlet Coorbit Spaces; 4.2 Shearlet Atomic Decompositions and Shearlet Banach Frames; 4.3 Nonlinear Approximation; 5 Structure of Shearlet Coorbit Spaces; 5.1 Atomic Decomposition of Besov Spaces; 5.2 A Density Result; 5.3 Traces on the Real Axes; 5.4 Embedding Results; 6 Analysis of Singularities; 6.1 Hyperplane Singularities; 6.2 Tetrahedron Singularities; References; Shearlets and Optimally Sparse Approximations; 1 Introduction; 1.1 Choice of Model for Anisotropic Features
1.2 Measure for Sparse Approximation and Optimality
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