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Birkhäuser

Quaternion and Clifford Fourier Transforms and Wavelets

  • Book
  • © 2013

Overview

Editors:
  1. Eckhard Hitzer

    1. , Department of Material Science, International Christian University, Tokyo, Japan

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    You can also search for this editor in PubMed   Google Scholar

  2. Stephen J. Sangwine

    1. School of Computer Science and, Electronic Engineering, University of Essex, Colchester, United Kingdom

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    You can also search for this editor in PubMed   Google Scholar

  • 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 book series: Trends in Mathematics (TM)

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-xxvii
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  2. Quaternions

    1. Front Matter

      Pages 1-1
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    2. Quaternion Fourier Transform: Re-tooling Image and Signal Processing Analysis

      • Todd Anthony Ell
      Pages 3-14

    3. The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations

      • Eckhard Hitzer, Stephen J. Sangwine
      Pages 15-39

    4. Quaternionic Spectral Analysis of Non-Stationary Improper Complex Signals

      • Nicolas Le Bihan, Stephen J. Sangwine
      Pages 41-56

    5. Quaternionic Local Phase for Low-level Image Processing Using Atomic Functions

      • E. Ulises Moya-Sánchez, E. Bayro-Corrochano
      Pages 57-83

    6. Bochner’s Theorems in the Framework of Quaternion Analysis

      • S. Georgiev, J. Morais
      Pages 85-104

    7. Bochner–Minlos Theorem and Quaternion Fourier Transform

      • S. Georgiev, J. Morais, K. I. Kou, W. Sprößig
      Pages 105-120

  3. Clifford Algebra

    1. Front Matter

      Pages 121-121
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    2. Square Roots of –1 in Real Clifford Algebras

      • Eckhard Hitzer, Jacques Helmstetter, RafaĹ‚ AbĹ‚amowicz
      Pages 123-153

    3. A General Geometric Fourier Transform

      • Roxana Bujack, Gerik Scheuermann, Eckhard Hitzer
      Pages 155-176

    4. Clifford–Fourier Transform and Spinor Representation of Images

      • Thomas Batard, Michel Berthier
      Pages 177-195

    5. Analytic Video (2D + t) Signals Using Clifford–Fourier Transforms in Multiquaternion Grassmann–Hamilton–Clifford Algebras

      • P. R. Girard, R. Pujol, P. Clarysse, A. Marion, R. Goutte, P. Delachartre
      Pages 197-219

    6. Generalized Analytic Signals in Image Processing: Comparison, Theory and Applications

      • Swanhild Bernstein, Jean-Luc Bouchot, Martin Reinhardt, Bettina Heise
      Pages 221-246

    7. Colour Extension of Monogenic Wavelets with Geometric Algebra: Application to Color Image Denoising

      • RaphaĂ«l Soulard, Philippe CarrĂ©
      Pages 247-268

    8. Seeing the Invisible and Maxwell’s Equations

      • Swanhild Bernstein
      Pages 269-284

    9. A Generalized Windowed Fourier Transform in Real Clifford Algebra Cl 0,n

      • Mawardi Bahri
      Pages 285-298

    10. The Balian–Low Theorem for the Windowed Clifford–Fourier Transform

      • Yingxiong Fu, Uwe Kähler, Paula Cerejeiras
      Pages 299-319

    11. Sparse Representation of Signals in Hardy Space

      • Shuang Li, Tao Qian
      Pages 321-332

  4. Back Matter

    Pages 333-338
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Keywords

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.

Editors and Affiliations

  • , Department of Material Science, International Christian University, Tokyo, Japan

    Eckhard Hitzer

  • School of Computer Science and, Electronic Engineering, University of Essex, Colchester, United Kingdom

    Stephen J. Sangwine

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