Trends in Mathematics, 2013, Quaternion and Clifford Fourier Transforms and Wavelets, Editors: Eckhard Hitzer, Stephen J. Sangwine, ISBN: 978-3-0348-0602-2 (Print) 978-3-0348-0603-9 (Online), 366 pages, 101.60 Euros (Amazon), preface and index, Birkhauser, Basel, 2013.
Short description: 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.
Table of contents (16 chapters)
- Front Matter including QCFTW History Chapter (by F. Brackx, E. Hitzer, S. Sangwine), Pages i-xxvii, Free Download PDF (509KB)
- Front Matter, Pages 1-1, Free Download PDF (17KB)
- Pages 3-14Quaternion Fourier Transform: Re-tooling Image and Signal Processing Analysis
- Pages 15-39The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations
- Pages 41-56Quaternionic Spectral Analysis of Non-Stationary Improper Complex Signals
- Pages 57-83Quaternionic Local Phase for Low-level Image Processing Using Atomic Functions
- Pages 85-104Bochner’s Theorems in the Framework of Quaternion Analysis
- Pages 105-120Bochner–Minlos Theorem and Quaternion Fourier Transform
- Clifford Algebra
- Front Matter, Pages 121-121, Download PDF (17KB)
- Pages 123-153Square Roots of –1 in Real Clifford Algebras
- Pages 155-176A General Geometric Fourier Transform
- Pages 177-195Clifford–Fourier Transform and Spinor Representation of Images
- Pages 197-219Analytic Video (2D + t) Signals Using Clifford–Fourier Transforms in Multiquaternion Grassmann–Hamilton–Clifford Algebras
- Pages 221-246Generalized Analytic Signals in Image Processing: Comparison, Theory and Applications
- Pages 247-268Colour Extension of Monogenic Wavelets with Geometric Algebra: Application to Color Image Denoising
- Pages 269-284Seeing the Invisible and Maxwell’s Equations
- Pages 285-298A Generalized Windowed Fourier Transform in Real Clifford Algebra Cl 0,n
- Pages 299-319The Balian–Low Theorem for the Windowed Clifford–Fourier Transform
- Pages 321-332Sparse Representation of Signals in Hardy Space
- Back Matter, Pages 333-338, Free Download PDF (113KB)