Quaternion and Clifford Fourier Transforms

Description

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, electric 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. Cover Half Title Title Page Copyright Page Contents Foreword Preface Acknowlegements Chapter 1: Introduction 1.1. BRIEF HISTORICAL NOTES 1.1.1. Definitions related to Clifford’s geometric algebras 1.2. QUATERNION FOURIER TRANSFORMS (QFT) 1.2.1. Major developments in the history of the quaternion Fourier transform 1.2.2. Towards splitting quaternions and the QFT 1.3. DEVELOPMENT OF CLIFFORD FOURIER TRANSFORMATIONS IN CLIFFORD’S GEOMETRIC ALGEBRA 1.3.1. On the role of Clifford algebra square roots of -1 for Clifford Fourier transformations 1.3.2. Clifford Fourier transform in the light of Clifford analysis 1.4. DEVELOPMENT OF QUATERNION AND CLIFFORD WAVELETS 1.4.1. Towards Clifford wavelets in Clifford analysis 1.4.2. Further developments in quaternion and Clifford wavelet theory Chapter 2: Clifford algebra 2.1. AXIOMS OF CLIFFORD’S GEOMETRIC ALGEBRA 2.1.1. Axioms for geometric algebra Rp,q = Cl(p, q) 2.1.1.1. Algebra over field of real numbers 2.1.1.2. Definition (1): using quadratic form 2.1.1.3. Definition (2): by basic multiplication rules 2.1.1.4. Definition (3): compact form 2.1.1.5. Grade r subspaces 2.1.1.6. Reverse and principal reverse 2.1.2. Geometric algebra R2 = Cl(2, 0) 2.1.2.1. Complex numbers 2.1.2.2. Reflections and rotations 2.1.2.3. Two-dimensional point groups 2.1.2.4. Clifford’s geometric algebra G3 = Cl(3, 0) of R3 2.2. QUADRATIC FORMS IN CLIFFORD’S GEOMETRIC ALGEBRA 2.2.1. Definition of geometric algebra with quadratic form 2.2.2. Examples of quadratic forms and associated geometric algebras 2.2.3. Geometric algebras with quadratic form of signature p = q 2.2.3.1. A new interpretation of the geometric algebra of the Minkowski plane 2.2.3.2. Generalizing to the geometric mother algebra with p = q = n 2.3. CLIFFORD’S GEOMETRIC PRODUCT AND DERIVED PRODUCTS 2.3.1. The geometric product of multivectors 2.3.2. The scalar product 2.3.3. The outer product 2.3.3.1. The cross product of three dimensions 2.3.3.2. Linear dependence and independence 2.3.4. Right and left contraction 2.4. DETERMINANTS IN GEOMETRIC ALGEBRA 2.4.1. Determinant definition 2.4.2. Adjoint and inverse linear maps 2.5. GRAM-SCHMIDT ORTHOGONALIZATION IN GEOMETRIC ALGEBRA 2.6. IMPORTANT CLIFFORD GEOMETRIC ALGEBRAS 2.6.1. Overview of Clifford’s geometric algebra of the Euclidean plane 2.6.2. Example of Cl(2, 0) 2.6.2.1. Aspects of algebraic unification and vector inverse 2.6.2.2. On geometric operations and transformations 2.6.2.3. Concepts of vectors, k-vectors and multivectors 2.6.2.4. Higher dimensional types of inner and outer products for multivectors 2.6.3. Overview of Clifford’s geometric algebra of the 3D Euclidean space 2.6.3.1. Explicit multiplication table and important subalgebras of Cl(3, 0) 2.6.3.2. Concepts of grade structure of Cl(3, 0) and duality 2.6.3.3. Concept of blade subspaces of Cl(3, 0) 2.6.3.4. Useful algebraic formulas in Clifford’s geometric algebras 2.6.4. Extending Clifford’s geometric algebra to spacetime (STA) 2.6.5. Geometric modeling with conformal extension 2.6.5.1. On points, planes and motors in Cl(4, 1) 2.6.5.2. Modeling geometric objects in Cl(4, 1) with blades 2.6.6. On Clifford analysis 2.7. HOW IMAGINARY NUMBERS BECOME REAL IN CLIFFORD ALGEBRAS 2.7.1. What is an imaginary number? 2.7.2. What about imaginary eigenvalues and complex eigenvectors? 2.7.2.1. Two real dimensions 2.7.2.1.1. Complex treatment 2.7.2.1.2. Real explanation 2.7.2.2. Three real dimensions 2.7.2.2.1. Complex treatment of three dimensions 2.7.2.2.2. Real explanation for three dimensions 2.7.3. From imaginary numbers to multivector square roots of -1 in Clifford geometric algebras Cl(p, q) with p + q