© 2013

Quaternion and Clifford Fourier Transforms and Wavelets

  • Eckhard Hitzer
  • Stephen J. Sangwine


  • First book on this topics combining theory and applications

  • Comprehensive overview addressing both mathematicians and engineers

  • Provides a representative overview of the relevant literature


Part of the Trends in Mathematics book series (TM)

Table of contents

  1. Front Matter
    Pages i-xxvii
  2. Quaternions

    1. Front Matter
      Pages 1-1
    2. Nicolas Le Bihan, Stephen J. Sangwine
      Pages 41-56
    3. E. Ulises Moya-Sánchez, E. Bayro-Corrochano
      Pages 57-83
    4. S. Georgiev, J. Morais, K. I. Kou, W. Sprößig
      Pages 105-120
  3. Clifford Algebra

    1. Front Matter
      Pages 121-121
    2. Eckhard Hitzer, Jacques Helmstetter, Rafał Abłamowicz
      Pages 123-153
    3. Roxana Bujack, Gerik Scheuermann, Eckhard Hitzer
      Pages 155-176
    4. Thomas Batard, Michel Berthier
      Pages 177-195
    5. P. R. Girard, R. Pujol, P. Clarysse, A. Marion, R. Goutte, P. Delachartre
      Pages 197-219
    6. Swanhild Bernstein, Jean-Luc Bouchot, Martin Reinhardt, Bettina Heise
      Pages 221-246
    7. Swanhild Bernstein
      Pages 269-284
    8. Yingxiong Fu, Uwe Kähler, Paula Cerejeiras
      Pages 299-319
    9. Shuang Li, Tao Qian
      Pages 321-332
  4. Back Matter
    Pages 333-338

About this book


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.


complex numbers hypercomplex algebra and analysis image and signal processing matrix isomorphisms quaternionic spectral analysis

Editors and affiliations

  • Eckhard Hitzer
    • 1
  • Stephen J. Sangwine
    • 2
  1. 1., Department of Material ScienceInternational Christian UniversityTokyoJapan
  2. 2.School of Computer Science and, Electronic EngineeringUniversity of EssexColchesterUnited Kingdom

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