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
Part of the Trends in Mathematics book series (TM)


We present an algebraic framework for (2D+ t) video analytic signals and a numerical implementation thereof using Clifford biquaternions and Clifford–Fourier transforms. Though the basic concepts of Clifford–Fourier transforms are well known, an implementation of analytic video sequences using multiquaternion algebras does not seem to have been realized so far. After a short presentation of multiquaternion Clifford algebras and Clifford–Fourier transforms, a brief pedagogical review of 1D and 2D quaternion analytic signals using right quaternion Fourier transforms is given. Then, the biquaternion algebraic framework is developed to express Clifford–Fourier transforms and (2D + t) video analytic signals in standard and polar form constituted by a scalar, a pseudoscalar and six phases. The phase extraction procedure is fully detailed. Finally, a numerical implementation using discrete fast Fourier transforms of an analytic multiquaternion video signal is provided.


Multiquaternion Clifford algebras Clifford biquaternions Clifford–Fourier transforms 2D + t analytic video signals. 


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Copyright information

© Springer Basel 2013

Authors and Affiliations

  • P. R. Girard
    • 1
  • R. Pujol
    • 2
  • P. Clarysse
    • 1
  • A. Marion
    • 1
  • R. Goutte
    • 1
  • P. Delachartre
    • 1
  1. 1.CREATIS; CNRS UMR 5220 Inserm U1044; INSA-Lyon; Université Lyon 1, France Bât. Blaise PascalUniversité de LyonVilleurbanneFrance
  2. 2.Pôle de Mathématiques, INSA-Lyon Bât. Léonard de VinciUniversité de LyonVilleurbanneFrance

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