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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.

Harmonic analysis. Fourier analysis --- Numerical analysis --- Mathematics --- Information systems --- Computer. Automation --- gegevensopslag --- beeldverwerking --- Fourieranalyse --- toegepaste wiskunde --- numerieke analyse --- signaalverwerking --- Wavelets (Mathematics). --- Multivariate analysis. --- Fourier analysis. --- Signal processing. --- Image processing. --- Speech processing systems. --- Numerical analysis. --- Data structures (Computer science). --- Applied mathematics. --- Engineering mathematics. --- Fourier Analysis. --- Signal, Image and Speech Processing. --- Numerical Analysis. --- Data Storage Representation. --- Applications of Mathematics. --- Engineering --- Engineering analysis --- Mathematical analysis --- Information structures (Computer science) --- Structures, Data (Computer science) --- Structures, Information (Computer science) --- Electronic data processing --- File organization (Computer science) --- Abstract data types (Computer science) --- Computational linguistics --- Electronic systems --- Information theory --- Modulation theory --- Oral communication --- Speech --- Telecommunication --- Singing voice synthesizers --- Pictorial data processing --- Picture processing --- Processing, Image --- Imaging systems --- Optical data processing --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication) --- Analysis, Fourier

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In wavelet analysis, irregular wavelet frames have recently come to the forefront of current research due to questions concerning the robustness and stability of wavelet algorithms. A major difficulty in the study of these systems is the highly sensitive interplay between geometric properties of a sequence of time-scale indices and frame properties of the associated wavelet systems. This volume provides the first thorough and comprehensive treatment of irregular wavelet frames by introducing and employing a new notion of affine density as a highly effective tool for examining the geometry of sequences of time-scale indices. Many of the results are new and published for the first time. Topics include: qualitative and quantitative density conditions for existence of irregular wavelet frames, non-existence of irregular co-affine frames, the Nyquist phenomenon for wavelet systems, and approximation properties of irregular wavelet frames.

Wavelets (Mathematics) --- Ondelettes --- Electronic books. -- local. --- Harmonic analysis. --- Wavelets (Mathematics). --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Wavelet analysis --- Mathematics. --- Fourier analysis. --- Information theory. --- Fourier Analysis. --- Information and Communication, Circuits. --- Communication theory --- Communication --- Cybernetics --- Analysis, Fourier --- Mathematical analysis --- Math --- Science --- Banach algebras --- Calculus --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Harmonic analysis

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Hilbert space frames have long served as a valuable tool for signal and image processing due to their resilience to additive noise, quantization, and erasures, as well as their ability to capture valuable signal characteristics.  More recently, finite frame theory has grown into an important research topic in its own right, with a myriad of applications to pure and applied mathematics, engineering, computer science, and other areas.  The number of research publications, conferences, and workshops on this topic has increased dramatically over the past few years, but no survey paper or monograph has yet appeared on the subject. Edited by two of the leading experts in the field, Finite Frames aims to fill this void in the literature by providing a comprehensive, systematic study of finite frame theory and applications.  With carefully selected contributions written by highly experienced researchers, it covers topics including: * Finite Frame Constructions; * Optimal Erasure Resilient Frames; * Quantization of Finite Frames; * Finite Frames and Compressed Sensing; * Group and Gabor Frames; * Fusion Frames. Despite the variety of its chapters' source and content, the book's notation and terminology are unified throughout and provide a definitive picture of the current state of frame theory. With a broad range of applications and a clear, full presentation, this book is a highly valuable resource for graduate students and researchers across disciplines such as applied harmonic analysis, electrical engineering, quantum computing, medicine, and more.  It is designed to be used as a supplemental textbook, self-study guide, or reference book.

Mathematics --- Operator theory --- Harmonic analysis. Fourier analysis --- Computer science --- Artificial intelligence. Robotics. Simulation. Graphics --- Computer. Automation --- computervisie --- beeldverwerking --- Fourieranalyse --- analyse (wiskunde) --- toegepaste wiskunde --- grafische vormgeving --- informatica --- wiskunde --- KI (kunstmatige intelligentie) --- signaalverwerking --- Frames (Vector analysis). --- Finite frame theory. --- Mathematics. --- Fourier analysis. --- Computer vision. --- Operator theory. --- Approximations and Expansions. --- Signal, Image and Speech Processing. --- Fourier Analysis. --- Computer Imaging, Vision, Pattern Recognition and Graphics. --- Operator Theory. --- Applications of Mathematics. --- Functional analysis --- Machine vision --- Vision, Computer --- Artificial intelligence --- Image processing --- Pattern recognition systems --- Analysis, Fourier --- Mathematical analysis --- Math --- Science --- Approximation theory. --- Signal processing. --- Image processing. --- Speech processing systems. --- Optical data processing. --- Applied mathematics. --- Engineering mathematics. --- Computational linguistics --- Electronic systems --- Information theory --- Modulation theory --- Oral communication --- Speech --- Telecommunication --- Singing voice synthesizers --- Pictorial data processing --- Picture processing --- Processing, Image --- Imaging systems --- Optical data processing --- Engineering --- Engineering analysis --- Optical computing --- Visual data processing --- Bionics --- Electronic data processing --- Integrated optics --- Photonics --- Computers --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication) --- Theory of approximation --- Functions --- Polynomials --- Chebyshev systems --- Optical equipment --- Finite frame theory

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Hilbert space frames have long served as a valuable tool for signal and image processing due to their resilience to additive noise, quantization, and erasures, as well as their ability to capture valuable signal characteristics.  More recently, finite frame theory has grown into an important research topic in its own right, with a myriad of applications to pure and applied mathematics, engineering, computer science, and other areas.  The number of research publications, conferences, and workshops on this topic has increased dramatically over the past few years, but no survey paper or monograph has yet appeared on the subject. Edited by two of the leading experts in the field, Finite Frames aims to fill this void in the literature by providing a comprehensive, systematic study of finite frame theory and applications.  With carefully selected contributions written by highly experienced researchers, it covers topics including: * Finite Frame Constructions; * Optimal Erasure Resilient Frames; * Quantization of Finite Frames; * Finite Frames and Compressed Sensing; * Group and Gabor Frames; * Fusion Frames. Despite the variety of its chapters' source and content, the book's notation and terminology are unified throughout and provide a definitive picture of the current state of frame theory. With a broad range of applications and a clear, full presentation, this book is a highly valuable resource for graduate students and researchers across disciplines such as applied harmonic analysis, electrical engineering, quantum computing, medicine, and more.  It is designed to be used as a supplemental textbook, self-study guide, or reference book.

Mathematics --- Operator theory --- Harmonic analysis. Fourier analysis --- Mathematics --- Computer science --- Artificial intelligence. Robotics. Simulation. Graphics --- Computer. Automation --- computervisie --- beeldverwerking --- Fourieranalyse --- analyse (wiskunde) --- toegepaste wiskunde --- grafische vormgeving --- informatica --- wiskunde --- KI (kunstmatige intelligentie) --- signaalverwerking