Fast complexified quaternion Fourier transform

IEEE Transactions on Signal Processing

Volumn 56, Issue 4, 2008, Pages 1522-1531

  Said, Salem ^a   Le Bihan, Nicolas ^a   Sangwine, Stephen J ^b  

^a GIPSA LAB   (France)

^b UNIVERSITY OF ESSEX   (United Kingdom)

Author keywords

Biquaternion Fourier transform (BiQFT); Biquaternion valued signals; Fast algorithm (BiQFFT); Hermitian symmetries; Hyperanalytic signal

Indexed keywords

ALGORITHMS; COMPUTATIONAL METHODS;

BIQUATERNION FOURIER TRANSFORM; BIQUATERNION-VALUED SIGNALS; FAST ALGORITHM; HERMITIAN SYMMETRIES; HYPERANALYTIC SIGNAL;

FOURIER TRANSFORMS;

EID: 41849148997     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2007.910477     Document Type: Article

References (21)

1. S. J. Sangwine, "Fourier transforms of colour images using quaternions, or hypercomplex, numbers," Electron. Lett., vol. 32, no. 21, pp. 1979-1980, 1996.
2. S. Sangwine and N. LeBihan, Quaternion Toolbox for Matlab [Online]. Available: Http://qtfm.sourceforge.net/ (online software library).
3. T. A. Ell, "Hypercomplex spectral transformations," Ph.D. dissertation, University of Minnesota, Minneapolis, 1992.
4. V. M. Chernov, "Arithmetic method in the theory of discrete orthogonal transforms," Proc. SPIE, vol. 2363, pp. 134-141, 1995.
5. V. M. Chernov, "Discrete orthogonal transforms with data representation in composition algebras," in Proc. 9th Scandinavian Conf Image Analysis (SCIA), Uppsala, Sweden, 1995, vol. 1, pp. 357-364.
6. V. M. Chernov, "Algorithms of two-dimensional discrete orthogonal transforms realized in Hamilton-Eisenstein codes," Transl.:PRoblemy Peredachi Informatsii Prob. Inf. Transmission, vol. 31, no. 3, pp. 228-235, 1995.
7. G. Sommer, Ed., Geometric Computing With Clifford Algebras. New York: Springer, 2001.
8. S.-C. Pei and C.-M. Cheng, "A novel block truncation coding of color images by using quaternion-moment preserving principle," in Proc. IEEE Int. Symp. Circuits systems, Atlanta, GA, May 12-15, 1996, vol. 2, pp. 684-687.
9. T. Bülow and G. Sommer, "Hypercomplex signals - A novel extension of the analytic signal to the multidimensional case," IEEE Trans. Signal Process., vol. 49, no. 11, pp. 2844-2852, Nov. 2001.
10. M. Felsberg and G. Sommer, "The monogenic signal," IEEE Trans. Signal Process., vol. 49, no. 12, pp. 3136-3144, Dec. 2001.
11. J. Ebling and G. Scheuermann, "Clifford Fourier transform on vector fields," IEEE Trans. Vis. Comput. Graphics, vol. 11, no. 4, pp. 469-479, Jul.-Aug. 2005.
12. T. A. Ell and S. J. Sangwine, "Hypercomplex Fourier transforms of color images," IEEE Trans. Image Process., vol. 16, no. 1, pp. 22-35, Jan. 2007.
13. W. R. Hamilton, "On the geometrical interpretation of some results obtained by calculation with biquaternions," Proc. Roy. Irish Acad. vol. V, pp. 388-390, 1853.
14. J. P. Ward, Quaternions and Cayley Numbers, Algebra and Applications Norwell, MA: Kluwer Academic, 1997.
15. S. J. Sangwine, "Biquaternion (complexified quaternion) roots of-1," Adv. Appl. Clifford Alg., vol. 16, no. 1, pp. 63-68, 2006.
16. T. A. Ell and S. J. Sangwine, "Decomposition of 2-D hypercomplex Fourier transforms into pairs of complex Fourier transforms," in Proc. Eur. Conf. Signal Image Processing (EUSIPCO), Tampere, Finland, 2000, pp. 1061-1064.
17. M. Reimer and D. Yevick, "A clifford algebra analysis of polarization-mode dispersion and polarization-dependent loss," IEEE Photon. Technol. Lett., vol. 18, no. 6, pp. 734-736, 2006.
18. T. A. Ell and S. J. Sangwine, "Hypercomplex Wiener-Kintchine theorem with application to color image processing," in Proc. IEEE Int. Conf. Image Processing (ICIP), Vancouver, Canada, 2000, pp. 792-795.
19. D. Gabor, "Theory of communication," J. Inst. Elec. Eng., vol. 93, no. 26, pt. 111, pp. 429-457, 1946.
20. J. Ville, "Théorie et applications de la notion de signal analytique," Cables et Transmission, vol. 2A, pp. 61-74, 1948.
21. R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. Boston, MA: McGraw-Hill, 2000.