Advances in Applied Clifford Algebras

, Volume 27, Issue 1, pp 381–395 | Cite as

General two-sided quaternion Fourier transform, convolution and Mustard convolution



In this paper we use the general two-sided quaternion Fourier transform (QFT), and relate the classical convolution of quaternion-valued signals over \({{\mathbb R}^2}\) with the Mustard convolution. A Mustard convolution can be expressed in the spectral domain as the point wise product of the QFTs of the factor functions. In full generality do we express the classical convolution of quaternion signals in terms of finite linear combinations of Mustard convolutions, and vice versa the Mustard convolution of quaternion signals in terms of finite linear combinations of classical convolutions.


Convolution Mustard convolution Two-sided quaternion Fourier transform Quaternion signals Spatial domain Frequency domain 


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  1. 1.
    Bas, P.; Le Bihan, N.; Chassery, J.M.: Color image watermarking using quaternion Fourier transform. In: Acoustics, Speech, and Signal Processing, 2003. Proceedings (ICASSP03). IEEE International Conference on vol. 3, pp. III-521 (2003)Google Scholar
  2. 2.
    Bayro-Corrochano E.: The theory and use of the quaternion wavelet transform. J. Math. Imaging Vis. 24, 19–35 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bayro-Corrochano E., Trujillo N., Naranjo M.: Quaternion Fourier descriptors for the preprocessing and recognition of spoken words using images of spatiotemporal representations. J. Math. Imaging Vis. 28, 179–190 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bülow, T.: Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images. PhD Thesis, Univ. of Kiel (1999)Google Scholar
  5. 5.
    Bujack R., De Bie H., De Schepper N., Scheuermann G.: Convolution products for hypercomplex Fourier transforms. J. Math. Imaging Vis. 48, 606–624 preprint: (2014)
  6. 6.
    Coxeter H.S.M.: Quaternions and reflections. Am. Math. Mon. 53(3), 136–146 (1946)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    De Bie H., De Schepper N., Ell T.A., Rubrecht K., Sangwine S.J.: Connecting spatial and frequency domains for the quaternion Fourier transform. Appl. Math. Comput. 271, 581–593 (2015)MathSciNetGoogle Scholar
  8. 8.
    Denis P., Carré P., Fernandez-Maloigne C.: Spatial and spectral quaternionic approaches for colour images. Comput. Vis. Image Underst. 107, 74–87 (2007)CrossRefGoogle Scholar
  9. 9.
    Ell, TA.: Quaternionic-Fourier transform for analysis of two-dimensional linear time-invariant partial differential systems. In: Proceedings of the 32nd IEEE Conference on Decision and Control, December 15–17, vol. 2, pp. 1830–1841 (1993)Google Scholar
  10. 10.
    Ell, TA., Sangwine, SJ.: Hypercomplex Fourier transforms of color images. IEEE Trans. Image Process. 16(1), 22–35 (2007)Google Scholar
  11. 11.
    Guo C., Zhang L.: A novel multiresolution spatiotemporal saliency detection model and its applications in image and video compression. IEEE Trans. Image Process. 19, 185–198 (2010)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hildenbrand, D.: Foundations of Geometric Algebra Computing, Springer, Berlin (2013)Google Scholar
  13. 13.
    Hitzer, E.: Quaternion Fourier transform on quaternion fields and generalizations. Adv. Appl. Clifford Algebra 17, 497–517 (2007). doi: 10.1007/s00006-007-0037-8, preprint:
  14. 14.
    Hitzer, E.: Directional uncertainty principle for quaternion Fourier transforms, Adv. Appl. Clifford Algebra 20(2), 271–284 (2010). doi: 10.1007/s00006-009-0175-2, preprint:
  15. 15.
    Hitzer, E.: OPS-QFTs: A new type of quaternion Fourier transforms based on the orthogonal planes split with one or two general pure quaternions. In: Numerical Analysis and Applied Mathematics ICNAAM 2011, AIP Conf. Proc. 1389, pp. 280–283 (2011). doi: 10.1063/1.3636721, preprint:
  16. 16.
  17. 17.
    Hitzer, E.; Sangwine, SJ.: The orthogonal 2D planes split of quaternions and steerable quaternion Fourier transformations, in: E. Hitzer and S.J. Sangwine (Eds.), Quaternion and Clifford Fourier Transforms and Wavelets, Trends in Mathematics, vol. 27, pp. 15–40. Birkhäuser (2013) doi: 10.1007/978-3-0348-0603-9_2, preprint:
  18. 18.
    Hitzer, E., Helmstetter, J., Abłamowicz, R.: Square Roots of 1 in Real Clifford Algebras. In: Hitzer, E., Sangwine S.J. (eds.) Quaternion and Clifford Fourier Transforms and Wavelets, Trends in Mathematics, vol. 27, pp. 123–153. Birkhäuser (2013). doi: 10.1007/978-3-0348-0603-9_7, preprint:
  19. 19.
    Hitzer, E.: Two-Sided Clifford Fourier Transform with Two Square Roots of 1 in Cl(p, q). Adv. Appl. Clifford Algebras 24, 313–332 (2014). doi: 10.1007/s00006-014-0441-9, preprint:
  20. 20.
    Hitzer, E.: The orthogonal planes split of quaternions and its relation to quaternion geometry of rotations. IOP J. Phys. Conf. Ser. (JPCS), 597, 012042 (2015). doi: 10.1088/1742-6596/597/1/012042. Open Access URL:, preprint:
  21. 21.
    Jin L., Liu H., Xu X., Song E.: Quaternion-based impulse noise removal from color video sequences. IEEE Trans. Circ. Syst. Video Technol. 23, 741–755 (2013)CrossRefGoogle Scholar
  22. 22.
    Meister L., Schaeben H.: A concise quaternion geometry of rotations. Math. Meth. Appl. Sci. 28, 101–126 (2005)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Moxey C.E., Sangwine S.J., Ell T.A.: Hypercomplex correlation techniques for vector images. IEEE Trans. Signal Process. 51, 1941–1953 (2003)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Mustard D.: Fractional convolution. J. Aust. Math. Soc. Ser. B Vol. 40, 257–265 (1998)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Sangwine S.J.: Color image edge detector based on quaternion convolution. Electron. Lett. 34, 969–971 (1998)CrossRefGoogle Scholar
  26. 26.
    Sangwine S.J.: Fourier transforms of colour images using quaternion, or hypercomplex, numbers. Electron. Lett. 32(21), 1979–1980 (1996)CrossRefGoogle Scholar
  27. 27.
    Sangwine, S.J., Le Bihan, N.: Quaternion and octonion toolbox for Matla. Accessed 29 Mar 2016
  28. 28.
    Soulard R., Carré P.: Quaternionic wavelets for texture classification. Pattern Recognit. Lett. 32, 1669–1678 (2011)CrossRefGoogle Scholar

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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Mitaka, TokyoJapan

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