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Quaternion And Clifford Fourier Transforms
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Book Synopsis Quaternion and Clifford Fourier Transforms and Wavelets by : Eckhard Hitzer
Download or read book Quaternion and Clifford Fourier Transforms and Wavelets written by Eckhard Hitzer and published by Springer Science & Business Media. This book was released on 2013-06-24 with total page 358 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quaternion and Clifford Fourier and wavelet transformations generalize the classical theory to higher dimensions and are becoming increasingly important in diverse areas of mathematics, physics, computer science and engineering. This edited volume presents the state of the art in these hypercomplex transformations. The Clifford algebras unify Hamilton’s quaternions with Grassmann algebra. A Clifford algebra is a complete algebra of a vector space and all its subspaces including the measurement of volumes and dihedral angles between any pair of subspaces. Quaternion and Clifford algebras permit the systematic generalization of many known concepts. This book provides comprehensive insights into current developments and applications including their performance and evaluation. Mathematically, it indicates where further investigation is required. For instance, attention is drawn to the matrix isomorphisms for hypercomplex algebras, which will help readers to see that software implementations are within our grasp. It also contributes to a growing unification of ideas and notation across the expanding field of hypercomplex transforms and wavelets. The first chapter provides a historical background and an overview of the relevant literature, and shows how the contributions that follow relate to each other and to prior work. The book will be a valuable resource for graduate students as well as for scientists and engineers.
Book Synopsis Quaternion and Clifford Fourier Transforms by : Eckhard Hitzer
Download or read book Quaternion and Clifford Fourier Transforms written by Eckhard Hitzer and published by CRC Press. This book was released on 2021-09-21 with total page 475 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quaternion and Clifford Fourier Transforms describes the development of quaternion and Clifford Fourier transforms in Clifford (geometric) algebra over the last 30 years. It is the first comprehensive, self-contained book covering this vibrant new area of pure and applied mathematics in depth. The book begins with a historic overview, followed by chapters on Clifford and quaternion algebra and geometric (vector) differential calculus (part of Clifford analysis). The core of the book consists of one chapter on quaternion Fourier transforms and one on Clifford Fourier transforms. These core chapters and their sections on more special topics are reasonably self-contained, so that readers already somewhat familiar with quaternions and Clifford algebra will hopefully be able to begin reading directly in the chapter and section of their particular interest, without frequently needing to skip back and forth. The topics covered are of fundamental interest to pure and applied mathematicians, physicists, and engineers (signal and color image processing, electrical engineering, computer science, computer graphics, artificial intelligence, geographic information science, aero-space engineering, navigation, etc.). Features Intuitive real geometric approach to higher-dimensional Fourier transformations A comprehensive reference, suitable for graduate students and researchers Includes detailed definitions, properties, and many full step-by-step proofs Many figures and tables, a comprehensive biography, and a detailed index make it easy to locate information
Book Synopsis Quaternion Fourier Transforms for Signal and Image Processing by : Todd A. Ell
Download or read book Quaternion Fourier Transforms for Signal and Image Processing written by Todd A. Ell and published by John Wiley & Sons. This book was released on 2014-06-23 with total page 168 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on updates to signal and image processing technology made in the last two decades, this text examines the most recent research results pertaining to Quaternion Fourier Transforms. QFT is a central component of processing color images and complex valued signals. The book’s attention to mathematical concepts, imaging applications, and Matlab compatibility render it an irreplaceable resource for students, scientists, researchers, and engineers.
Book Synopsis Geometric Computing with Clifford Algebras by : Gerald Sommer
Download or read book Geometric Computing with Clifford Algebras written by Gerald Sommer and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 559 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph-like anthology introduces the concepts and framework of Clifford algebra. It provides a rich source of examples of how to work with this formalism. Clifford or geometric algebra shows strong unifying aspects and turned out in the 1960s to be a most adequate formalism for describing different geometry-related algebraic systems as specializations of one "mother algebra" in various subfields of physics and engineering. Recent work shows that Clifford algebra provides a universal and powerful algebraic framework for an elegant and coherent representation of various problems occurring in computer science, signal processing, neural computing, image processing, pattern recognition, computer vision, and robotics.
Book Synopsis Visualizing Quaternions by : Andrew J. Hanson
Download or read book Visualizing Quaternions written by Andrew J. Hanson and published by Elsevier. This book was released on 2006-02-06 with total page 530 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduced 160 years ago as an attempt to generalize complex numbers to higher dimensions, quaternions are now recognized as one of the most important concepts in modern computer graphics. They offer a powerful way to represent rotations and compared to rotation matrices they use less memory, compose faster, and are naturally suited for efficient interpolation of rotations. Despite this, many practitioners have avoided quaternions because of the mathematics used to understand them, hoping that some day a more intuitive description will be available.The wait is over. Andrew Hanson's new book is a fresh perspective on quaternions. The first part of the book focuses on visualizing quaternions to provide the intuition necessary to use them, and includes many illustrative examples to motivate why they are important—a beautiful introduction to those wanting to explore quaternions unencumbered by their mathematical aspects. The second part covers the all-important advanced applications, including quaternion curves, surfaces, and volumes. Finally, for those wanting the full story of the mathematics behind quaternions, there is a gentle introduction to their four-dimensional nature and to Clifford Algebras, the all-encompassing framework for vectors and quaternions. - Richly illustrated introduction for the developer, scientist, engineer, or student in computer graphics, visualization, or entertainment computing. - Covers both non-mathematical and mathematical approaches to quaternions.
Book Synopsis Quaternionic Integral Transforms by : Eckhard Hitzer
Download or read book Quaternionic Integral Transforms written by Eckhard Hitzer and published by Springer Nature. This book was released on 2023-09-09 with total page 187 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a machine-generated literature overview of quaternion integral transforms from select papers published by Springer Nature, which have been organized and introduced by the book’s editor. Each chapter presents summaries of predefined themes and provides the reader with a basis for further exploration of the topic. As one of the experimental projects initiated by Springer Nature for AI book content generation, this book shows the latest developments in the field. It will be a useful reference for students and researchers who are interested in exploring the latest developments in quaternion integral transforms.
Book Synopsis Towards Intelligent Systems Modeling and Simulation by : Samsul Ariffin Abdul Karim
Download or read book Towards Intelligent Systems Modeling and Simulation written by Samsul Ariffin Abdul Karim and published by Springer Nature. This book was released on 2021-09-17 with total page 439 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book creates the emergence of disruptive technologies that have led to a significant change in the role of mathematics and statistics for problem solving, with the use of sophisticated software and hardware in solving complex systems and process. In the era of digital technology, mathematics and statistics need to be highly relevant to be able to cater for the needs of IR4.0 such as big data analytics, simulation, autonomous system, and cloud computing. Motivated by this development, a total of 26 chapters are contributed by respectable experts for this book. The main scope of the book is to conduct a new system of modeling and simulations on solving differential equations, nonlinear equations, energy, epidemiology, and risk assessment. This book is of interest for postgraduate students, researchers as well as other scientists who are working in numerical modeling and simulations based on efficient mathematical and statistical techniques.
Book Synopsis Quaternion Algebras by : John Voight
Download or read book Quaternion Algebras written by John Voight and published by Springer Nature. This book was released on 2021-06-28 with total page 877 pages. Available in PDF, EPUB and Kindle. Book excerpt: This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout.
Download or read book Fourier and Laplace Transforms written by and published by Cambridge University Press. This book was released on 2003-08-07 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook presents in a unified manner the fundamentals of both continuous and discrete versions of the Fourier and Laplace transforms. These transforms play an important role in the analysis of all kinds of physical phenomena. As a link between the various applications of these transforms the authors use the theory of signals and systems, as well as the theory of ordinary and partial differential equations. The book is divided into four major parts: periodic functions and Fourier series, non-periodic functions and the Fourier integral, switched-on signals and the Laplace transform, and finally the discrete versions of these transforms, in particular the Discrete Fourier Transform together with its fast implementation, and the z-transform. This textbook is designed for self-study. It includes many worked examples, together with more than 120 exercises, and will be of great value to undergraduates and graduate students in applied mathematics, electrical engineering, physics and computer science.
Book Synopsis Geometric Algebra Applications Vol. III by : Eduardo Bayro-Corrochano
Download or read book Geometric Algebra Applications Vol. III written by Eduardo Bayro-Corrochano and published by Springer Nature. This book was released on with total page 662 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Fourier Transforms by : Goran Nikolic
Download or read book Fourier Transforms written by Goran Nikolic and published by BoD – Books on Demand. This book was released on 2017-02-08 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main purpose of this book is to provide a modern review about recent advances in Fourier transforms as the most powerful analytical tool for high-tech application in electrical, electronic, and computer engineering, as well as Fourier transform spectral techniques with a wide range of biological, biomedical, biotechnological, pharmaceutical, and nanotechnological applications. The confluence of Fourier transform methods with high tech opens new opportunities for detection and handling of atoms and molecules using nanodevices, with potential for a large variety of scientific and technological applications.
Book Synopsis Clifford Analysis and Its Applications by : F. Brackx
Download or read book Clifford Analysis and Its Applications written by F. Brackx and published by Springer Science & Business Media. This book was released on 2001-07-31 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: In its traditional form, Clifford analysis provides the function theory for solutions of the Dirac equation. From the beginning, however, the theory was used and applied to problems in other fields of mathematics, numerical analysis, and mathematical physics. recently, the theory has enlarged its scope considerably by incorporating geometrical methods from global analysis on manifolds and methods from representation theory. New, interesting branches of the theory are based on conformally invariant, first-order systems other than the Dirac equation, or systems that are invariant with respect to a group other than the conformal group. This book represents an up-to-date review of Clifford analysis in its present form, its applications, and directions for future research. Readership: Mathematicians and theoretical physicists interested in Clifford analysis itself, or in its applications to other fields.
Book Synopsis Clifford Algebra to Geometric Calculus by : David Hestenes
Download or read book Clifford Algebra to Geometric Calculus written by David Hestenes and published by Springer Science & Business Media. This book was released on 1984 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt: Matrix algebra has been called "the arithmetic of higher mathematics" [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebra' suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quaternions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas.
Book Synopsis Linear Canonical Transforms by : John J. Healy
Download or read book Linear Canonical Transforms written by John J. Healy and published by Springer. This book was released on 2015-11-26 with total page 463 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a clear and accessible introduction to the essential mathematical foundations of linear canonical transforms from a signals and systems perspective. Substantial attention is devoted to how these transforms relate to optical systems and wave propagation. There is extensive coverage of sampling theory and fast algorithms for numerically approximating the family of transforms. Chapters on topics ranging from digital holography to speckle metrology provide a window on the wide range of applications. This volume will serve as a reference for researchers in the fields of image and signal processing, wave propagation, optical information processing and holography, optical system design and modeling, and quantum optics. It will be of use to graduate students in physics and engineering, as well as for scientists in other areas seeking to learn more about this important yet relatively unfamiliar class of integral transformations.
Book Synopsis Exploring physics with Geometric Algebra by : Peeter Joot
Download or read book Exploring physics with Geometric Algebra written by Peeter Joot and published by Peeter Joot. This book was released on with total page 1106 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an exploratory collection of notes containing worked examples of a number of applications of Geometric Algebra (GA), also known as Clifford Algebra. This writing is focused on undergraduate level physics concepts, with a target audience of somebody with an undergraduate engineering background (i.e. me at the time of writing.) These notes are more journal than book. You'll find lots of duplication, since I reworked some topics from scratch a number of times. In many places I was attempting to learn both the basic physics concepts as well as playing with how to express many of those concepts using GA formalisms. The page count proves that I did a very poor job of weeding out all the duplication. These notes are (dis)organized into the following chapters * Basics and Geometry. This chapter covers a hodge-podge collection of topics, including GA forms for traditional vector identities, Quaterions, Cauchy equations, Legendre polynomials, wedge product representation of a plane, bivector and trivector geometry, torque and more. A couple attempts at producing an introduction to GA concepts are included (none of which I was ever happy with.) * Projection. Here the concept of reciprocal frame vectors, using GA and traditional matrix formalisms is developed. Projection, rejection and Moore-Penrose (generalized inverse) operations are discussed. * Rotation. GA Rotors, Euler angles, spherical coordinates, blade exponentials, rotation generators, and infinitesimal rotations are all examined from a GA point of view. * Calculus. Here GA equivalents for a number of vector calculus relations are developed, spherical and hyperspherical volume parameterizations are derived, some questions about the structure of divergence and curl are examined, and tangent planes and normals in 3 and 4 dimensions are examined. Wrapping up this chapter is a complete GA formulation of the general Stokes theorem for curvilinear coordinates in Euclidean or non-Euclidean spaces is developed. * General Physics. This chapter introduces a bivector form of angular momentum (instead of a cross product), examines the components of radial velocity and acceleration, kinetic energy, symplectic structure, Newton's method, and a center of mass problem for a toroidal segment. * Relativity. This is a fairly incoherent chapter, including an attempt to develop the Lorentz transformation by requiring wave equation invariance, Lorentz transformation of the four-vector (STA) gradient, and a look at the relativistic doppler equation. * Electrodynamics. The GA formulation of Maxwell's equation (singular in GA) is developed here. Various basic topics of electrodynamics are examined using the GA toolbox, including the Biot-Savart law, the covariant form for Maxwell's equation (Space Time Algebra, or STA), four vectors and potentials, gauge invariance, TEM waves, and some Lienard-Wiechert problems. * Lorentz Force. Here the GA form of the Lorentz force equation and its relation to the usual vectorial representation is explored. This includes some application of boosts to the force equation to examine how it transforms under observe dependent conditions. * Electrodynamic stress energy. This chapter explores concepts of electrodynamic energy and momentum density and the GA representation of the Poynting vector and the stress-energy tensors. * Quantum Mechanics. This chapter includes a look at the Dirac Lagrangian, and how this can be cast into GA form. Properties of the Pauli and Dirac bases are explored, and how various matrix operations map onto their GA equivalents. A bivector form for the angular momentum operator is examined. A multivector form for the first few spherical harmonic eigenfunctions is developed. A multivector factorization of the three and four dimensional Laplacian and the angular momentum operators are derived. * Fourier treatments. Solutions to various PDE equations are attempted using Fourier series and transforms. Much of this chapter was exploring Fourier solutions to the GA form of Maxwell's equation, but a few other non-geometric algebra Fourier problems were also tackled.
Book Synopsis Empowering Novel Geometric Algebra for Graphics and Engineering by : Eckhard Hitzer
Download or read book Empowering Novel Geometric Algebra for Graphics and Engineering written by Eckhard Hitzer and published by Springer Nature. This book was released on 2023-04-26 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the proceedings of the Workshop Empowering Novel Geometric Algebra for Graphics and Engineering, ENGAGE 2022, held in conjunction with Computer Graphics International conference, CGI 2022, which took place virtually, in September 2022. The 10 full papers included in this volume were carefully reviewed and selected from 12 submissions. The workshop focused specifically on important aspects of geometric algebra including algebraic foundations, digitized transformations, orientation, conic fitting, protein modelling, digital twinning, and multidimensional signal processing.
Book Synopsis Clifford Algebras and Spinors by : Pertti Lounesto
Download or read book Clifford Algebras and Spinors written by Pertti Lounesto and published by Cambridge University Press. This book was released on 2001-05-03 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the second edition of a popular work offering a unique introduction to Clifford algebras and spinors. The beginning chapters could be read by undergraduates; vectors, complex numbers and quaternions are introduced with an eye on Clifford algebras. The next chapters will also interest physicists, and include treatments of the quantum mechanics of the electron, electromagnetism and special relativity with a flavour of Clifford algebras. This edition has three new chapters, including material on conformal invariance and a history of Clifford algebras.
