1st Edition

Quaternion and Clifford Fourier Transforms

By Eckhard Hitzer Copyright 2022
    474 Pages 31 B/W Illustrations
    by Chapman & Hall

    474 Pages 31 B/W Illustrations
    by Chapman & Hall

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    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.).


    • 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

    1. Introduction. 1.1. Brief Historical Notes. 1.2. Quaternion Fourier Transforms (QFT). 1.3. Clifford Fourier Transforms in Clifford’s Geometric Algebra. 1.4. Quaternion and Clifford Wavelets. 2. Clifford Algebra. 2.1. Axioms of Clifford Algebra. 2.2. Quadratic Forms in Clifford’s Geometric Algebra. 2.3. Clifford Product and Derived Products. 2.4. Determinants in Geometric Algebra. 2.5. Gram-Schmidt Orthogonalization in Geometric Algebra. 2.6. Important Clifford Geometric Algebras. 2.7. How Imaginary Numbers Become Real in Clifford Algebras. 2.8. Quaternions and Geometry of Rotations In 3, 4 Dim. 3. Geometric Calculus. 3.1. Introductory Notes on Vector Differential Calculus. 3.2. Brief Overview of Vector Differential Calculus. 3.3. Geometric Algebra for Differential Calculus. 3.4. Vector Differential Calculus. 3.5. Summary on Vector Differential Calculus. 4. Quaternion Fourier Transforms. 4.1. Fundamentals of Quaternion Fourier Transforms (QFT). 4.2. Properties of Quaternion Fourier Transform. 4.3. Special Quaternion Fourier Transforms. 4.4. From QFT To Volume-Time FT, Spacetime FT. 5. Clifford Fourier Transforms. 5.1. Overview of Clifford Fourier Transforms. 5.2. One-Sided Clifford Fourier Transforms. 5.3. Two-Sided Clifford Fourier Transforms. 5.4. Clifford Fourier Transform and Convolution. 5.5. Special Clifford Fourier Transforms. 6. Relating QFTs And CFTs. 6.1. Background. 6.2. General Geometric Fourier Transform. 6.3. CFT Due to Sommen And Bülow. 6.4. Color Image CFT. 6.5. Two-Sided CFT. 6.6. Quaternion Fourier Transform (QFT) 6.7. Quaternion Fourier Stieltjes Transform. 6.8. Quaternion Fourier Mellin Transform, Clifford Fourier Mellin Transform. 6.9. Volume-Time CFT and Spacetime CFT. 6.10. One-Sided CFTs. 6.11. Pseudoscalar Kernel CFTs. 6.12. Quaternion and Clifford Linear Canonical Transforms. 6.13. Summary Interrelationship Of QFTs, CFTs. Appendix A. Square Roots of −1 MAPLE, Cauchy-Schwarz, Uncertainty Equality. References. Bibliography. Index.


    Eckhard Hitzer has a PhD in theoretical physics from the University of Konstanz (Germany). He has been living in Japan since 1996 (Kyoto University, University of Fukui, and since 2012 as Senior Associate Professor at International Christian University [ICU] in Mitaka, Tokyo). He teaches Physics and Mathematics at ICU. He has published over 100 International Scientific journal papers and book chapters, is member of the editorial boards of three journals, author of one book, editor of two books and of 10 special journal issues and conference proceedings, active member and organizer of many scientific conference committees and prize committees. He edits the Email newsletter for everyone interested in Clifford Algebra and Geometric Algebra (GA-Net) since 2003, and the blog GA-Net Updates since 2012. He works on pure and applied Clifford (geometric) algebras, with specialization on space group symmetry in crystallography, neural network and artificial intelligence applications, and Clifford algebra based integral transformations. He has been co-organizing the session Quaternion and Clifford Fourier Transforms and Wavelets at the tri-annual International Conferences on Clifford Algebras and their Applications since past ten years, and the annual workshop Empowering Novel Geometric Algebra for Graphics & Engineering (ENGAGE) at the international conference Computer Graphics International (CGI) since past five years.