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Clifford Fourier transform on vector fields.

Julia Ebling1, Gerik Scheuermann

  • 1FB Informatik, University of Leipzig, 04315 Leipzig, Germany. ebling@informatik.uni-leipzig.de

IEEE Transactions on Visualization and Computer Graphics
|September 6, 2005
PubMed
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This study introduces the Clifford Fourier transform, extending Fourier analysis to vector fields and Clifford Algebra multivectors. This enables frequency analysis of vector fields and pattern matching for image processing applications.

Area of Science:

  • Image Processing
  • Computer Vision
  • Clifford Algebra

Background:

  • Established methods for scalar field analysis in image processing.
  • Need for robust feature extraction and theoretical basis for vector field analysis.

Purpose of the Study:

  • Extend Fourier transform to Clifford Algebra multivectors.
  • Develop frequency analysis for vector fields and vector-valued filters.
  • Enable pattern matching on vector fields using geometric properties.

Main Methods:

  • Introduction of Clifford convolution for unified scalar and vector field convolution.
  • Extension of the Fourier transform to Clifford Algebra elements (multivectors).
  • Development of Clifford convolution and derivative theorems.

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Main Results:

  • The Clifford Fourier transform enables frequency analysis of vector fields.
  • Vectors are transformed into multivectors in frequency space.
  • Basic vector patterns like sources, sinks, and vortices are described by multivectors.

Conclusions:

  • The Clifford Fourier transform provides a theoretical foundation for vector field analysis.
  • This method offers advantages for feature extraction and pattern matching in image processing.
  • Unified treatment of scalar and vector fields is achieved through Clifford Algebra.