Umfang:
Online-Ressource (XXVII, 338 p. 76 illus., 44 illus. in color, digital)
ISBN:
9783034806039
Serie:
Trends in Mathematics
Inhalt:
Preface.- History of Quaternion and Clifford-Fourier Transforms and Wavelets -- Part I: Quaternions.- 1 Quaternion Fourier Transform: Re-tooling Image and Signal Processing Analysis.- 2 The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations.- 3 Quaternionic Spectral Analysis of Non-Stationary Improper Complex Signals.- 4 Quaternionic Local Phase for Low-level Image Processing Using Atomic Functions.- 5 Bochner’s Theorems in the Framework of Quaternion Analysis.- 6 Bochner-Minlos Theorem and Quaternion Fourier Transform -- Part II: Clifford Algebra.- 7 Square Roots of -1 in Real Clifford Algebras.- 8 A General Geometric Fourier Transform.- 9 Clifford-Fourier Transform and Spinor Representation of Images -- 10 Analytic Video (2D+t) Signals Using Clifford-Fourier Transforms in Multiquaternion Grassmann-Hamilton-Clifford Algebras -- 11 Generalized Analytic Signals in Image Processing: Comparison, Theory and Applications -- 12 Color Extension of Monogenic Wavelets with Geometric Algebra: Application to Color Image Denoising -- 13 Seeing the Invisible and Maxwell’s Equations -- 14 A Generalized Windowed Fourier Transform in Real Clifford Algebra Cl_{0,n} -- 15 The Balian-Low theorem for the Windowed Clifford-Fourier Transform -- 16 Sparse Representation of Signals in Hardy Space. - Index.
Inhalt:
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.
Anmerkung:
Description based upon print version of record
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Contents; Preface; History of Quaternion and Clifford-Fourier Transforms and Wavelets; 1. Quaternion Fourier Transforms (QFT); 1.1. Major Developments in the History oft he Quaternion Fourier Transform; 1.2. Splitting Quaternions and the QFT; 2. Clifford-Fourier Transformations in Clifford's Geometric Algebra; 2.1. How Clifford Algebra Square Roots of - Lead to Clifford-Fourier Transformations; 2.2. The Clifford-Fourier Transform in the Light of Clifford Analysis; 3. Quaternion and Clifford Wavelets; 3.1. Clifford Wavelets in Clifford Analysis
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3.2. Further Developments in Quaternion and Clifford Wavelet TheoryReferences; Part I Quaternions; 1 Quaternion Fourier Transform: Re-tooling Image and Signal Processing Analysis; 1. Introduction; 2. Preliminaries; 2.1. Just the Facts; 2.2. Useful Subsets; 2.3. Useful Algebraic Equations; 3. Quaternion Fourier Transforms; 3.1. Transform Definitions; 3.2. Functional Relationships; 3.3. Relationships between Transforms; 4. Conclusions; Acknowledgment; References; 2 The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations; 1. Introduction
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2. Orthogonal Planes Split of Quaternions with Two Orthonormal Pure Unit Quaternions3. General Orthogonal 2D Planes Split; 3.1. Orthogonal 2D Planes Split Using Two Linearly Independent Pure Unit Quaternions; 3.2. Orthogonal 2D Planes Split Using One Pure Unit Quaternion; 3.3. Geometric Interpretation of Left and Right Exponential Factors in f, g; 3.4. Determination of f, g for Given Steerable Pair of Orthogonal 2D Planes; 4. New QFT Forms: OPS-QFTs with Two Pure Unit Quaternions f, g; 4.1. Generalized OPS Leads to New Steerable Type of QFT; 4.2. Two Phase Angle Version of QFT
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5. Involutions and QFTs Involving Quaternion Conjugation5.1. Involutions Involving Quaternion Conjugations; 5.2. New Steerable QFTs with Quaternion Conjugation and Two Pure Unit Quaternions f, g; 5.3. Local Geometric Interpretation ofthe QFT with Quaternion Conjugation; 5.4. Phase Angle QFT with Respect to f, g Including Quaternion Conjugation; 6. Conclusion; Acknowledgement; References; 3 Quaternionic Spectral Analysis of Non-Stationary Improper Complex Signals; 1. Introduction; 2. 1D Quaternion Fourier Transform; 2.1. Preliminary Remarks; 2.2. 1D Quaternion Fourier Transform
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2.3. Convolution2.4. The Quaternion Fourier Transform of the Hilbert Transform; 3. The Hypercomplex Analytic Signal; 4. Geometric Instantaneous Amplitude and Phase; 4.1. Complex Envelope; 4.2. Angular Velocity; 5. Conclusions; Acknowledgement; References; 4 Quaternionic Local Phase for Low-level Image Processing Using Atomic Functions; 1. Introduction; 2. Atomic Functions; 2.1. Mother Atomic Function up (x); 2.2. The dup(x) Function; 3. Why the Use of the Atomic Function?; 4. Quaternion Algebra H; 4.1. Quaternionic Atomic Function qup(x, y); 4.2. Quaternionic Atomic Wavelet
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4.3. Steerable Quaternionic Filter
Weitere Ausg.:
ISBN 9783034806022
Weitere Ausg.:
Erscheint auch als Druck-Ausgabe Quaternion and Clifford Fourier transforms and wavelets Basel : Birkhäuser, 2013 ISBN 9783034806022
Sprache:
Englisch
DOI:
10.1007/978-3-0348-0603-9
URL:
Volltext
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