Skip to main content
SpringerLink
Log in

Quaternionic Wiener–Khinchine Theorems and Spectral Representation of Convolution with Steerable Two-sided Quaternion Fourier Transform

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

In this paper we use the general two-sided quaternion Fourier transform (QFT) in order to derive Wiener–Khinchine theorems for the cross-correlation and for the auto-correlation of quaternion signals. Furthermore, we show how to derive a new four term spectral representation for the convolution of quaternion signals.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bas, P., Le Bihan, N., Chassery, J.M.: Color image watermarking using quaternion Fourier transform. In: Proceedings (ICASSP03) of 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 3, p. III-521 (2003)

  2. Bayro-Corrochano, E.: The theory and use of the quaternion wavelet transform. J. Math. Imag. Vis. 24, 19–35 (2006)

    Article  MathSciNet  Google Scholar 

  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. Imag. Vis. 28, 179–190 (2007)

    Article  MathSciNet  Google Scholar 

  4. Bülow, T.: Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images. Ph.D. Thesis, Univ. of Kiel (1999)

  5. Bujack, R., De Bie, H., De Schepper, N., Scheuermann, G.: Convolution products for hypercomplex Fourier transforms. J. Math. Imag. Vis. 48, 606–624 (2014, preprint). http://arxiv.org/abs/1303.1752

  6. Chládková, O.: EU-Türkei-Deal hilft der Demokratie nicht, ARD, Tagesschau of 15 June 2016. http://www.tagesschau.de/ausland/interview-duendar-101.html

  7. Coxeter, H.S.M.: Quaternions and reflections. Am. Math. Mon. 53(3), 136–146 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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)

    Article  MathSciNet  Google Scholar 

  9. Denis, P., Carré, P., Fernandez-Maloigne, C.: Spatial and spectral quaternionic approaches for colour images. Comput. Vis. Image Underst. 107, 74–87 (2007)

    Article  Google Scholar 

  10. Ell, T.A.: 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)

  11. Ell, T.A., Sangwine, S.J.: Hypercomplex Wiener–Khintchine theorem with application to color image correlation. Proc. Int. Conf. Image Process. 2, 792–795 (2000)

    Google Scholar 

  12. Ell, T.A., Sangwine, S.J.: Hypercomplex Fourier Transforms of Color Images. IEEE Transactions on Image Processing, vol. 16, no. 1 (2007)

  13. 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)

    Article  ADS  MathSciNet  Google Scholar 

  14. Hildenbrand, D.: Foundations of Geometric Algebra Computing. Springer, Berlin (2013)

    Book  MATH  Google Scholar 

  15. Hitzer, E.: Quaternion Fourier Transform on Quaternion Fields and Generalizations, Adv. in App. Cliff. Alg., vol. 17, pp. 497–517 (2007, preprint). doi:10.1007/s00006-007-0037-8, http://arxiv.org/abs/1306.1023

  16. Hitzer, E.: Directional uncertainty principle for quaternion Fourier transforms. Adv. Appl. Clifford Algebras 20(2), 271–284 (2010, preprint). doi:10.1007/s00006-009-0175-2, http://arxiv.org/abs/1306.1276

  17. 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., vol. 1389, pp. 280–283 (2011, preprint). doi:10.1063/1.3636721, http://arxiv.org/abs/1306.1650

  18. Hitzer, E.: Creative Peace License. http://gaupdate.wordpress.com/2011/12/14/the-creative-peace-license-14-dec-2011/

  19. Hitzer, E., Sangwine, S.J.: The orthogonal 2D planes split of quaternions and steerable quaternion Fourier transformations. In: Hitzer, E., Sangwine, S.J. (eds.) Quaternion and Clifford Fourier Transforms and Wavelets, Trends in Mathematics. Birkhäuser, pp. 15–40 (2013, preprint). doi:10.1007/978-3-0348-0603-9_2, http://arxiv.org/abs/1306.2157

  20. 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. Birkhäuser, pp. 123–153 (2013, preprint). doi:10.1007/978-3-0348-0603-9_7, http://arxiv.org/abs/1204.4576

  21. Hitzer, E.: Two-sided Clifford Fourier transform with two square roots of 1 in \(Cl(p, q)\) Adv. Appl. Clifford Algebras, vol. 24, pp. 313–332 (2014, preprint). doi:10.1007/s00006-014-0441-9, http://arxiv.org/abs/1306.2092

  22. 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, preprint). doi:10.1088/1742-6596/597/1/012042, http://iopscience.iop.org/1742-6596/597/1/012042/pdf/1742-6596_597_1_012042, http://vixra.org/abs/1411.0362

  23. Hitzer, E.: General two-sided quaternion Fourier transform, convolution and Mustard convolution. Adv. Appl. Clifford Algebras (preprint). doi:10.1007/s00006-016-0684-8, http://vixra.org/abs/1601.0165

  24. Jin, L., Liu, H., Xu, X., Song, E.: Quaternion-based impulse noise removal from color video sequences. IEEE Trans. Circuits Syst. Video Technol. 23, 741–755 (2013)

    Article  Google Scholar 

  25. Meister, L., Schaeben, H.: A concise quaternion geometry of rotations. Math. Methods Appl. Sci. 28, 101–126 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Moxey, C.E., Ell, T.A., Sangwine, S.J.: Hypercomplex operators and vector correlation, EUSIPCO, Eleventh European Signal Processing Conference, 3–6 September 2002, Toulouse, vol. III, pp. 247–250 (2002)

  27. Moxey, C.E., Sangwine, S.J., Ell, T.A.: Hypercomplex correlation techniques for vector images. IEEE Trans. Signal Process 51, 1941–1953 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  28. Sangwine, S.J.: Color image edge detector based on quaternion convolution. Electron. Lett. 34, 969–971 (1998)

    Article  Google Scholar 

  29. Sangwine, S.J.: Fourier transforms of colour images using quaternion, or hypercomplex, numbers. Electron. Lett. 32(21), 1979–1980 (1996)

    Article  Google Scholar 

  30. Sangwine, S.J., Le Bihan, N.: Quaternion and octonion toolbox for Matlab. http://qtfm.sourceforge.net/ (accessed 29 March 2016)

  31. Soulard, R., Carré, P.: Quaternionic wavelets for texture classification. Pattern Recognit. Lett. 32, 1669–1678 (2011)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Liberal Arts International Christian University, 181-8585, Mitaka, Tokyo, Japan

    Eckhard Hitzer

Authors
  1. Eckhard Hitzer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eckhard Hitzer.

Additional information

Communicated by Wolfgang Sprössig.

Soli Deo Gloria. Dedicated to Can Dündar, editor-in-chief of the Turkish newspaper Cumhuriyet, sentenced to five years and ten months in prison under Turkish “anti-terror” laws [6]. The use of this paper is subject to the Creative Peace License [18].

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hitzer, E. Quaternionic Wiener–Khinchine Theorems and Spectral Representation of Convolution with Steerable Two-sided Quaternion Fourier Transform. Adv. Appl. Clifford Algebras 27, 1313–1328 (2017). https://doi.org/10.1007/s00006-016-0744-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00006-016-0744-0

Keywords

Search

Navigation