Hypercomplex Fourier transforms of color images

IEEE Transactions on Image Processing

Volumn 16, Issue 1, 2007, Pages 22-35

  Ell, Todd A ^a   Sangwine, Stephen J ^b  

^a Goodrich Sensor Systems   (United States)

^b UNIVERSITY OF ESSEX   (United Kingdom)

Author keywords

Color image processing; Convolution; Correlation; Fourier transforms; Hypercomplex numbers; Quaternions

Indexed keywords

COLOR COMPUTER GRAPHICS; CONVOLUTION; CORRELATION THEORY; FOURIER TRANSFORMS; SIGNAL PROCESSING;

HYPERCOMPLEX NUMBERS; QUATERNIONS;

COLOR IMAGE PROCESSING;

ALGORITHM; ARTICLE; COLOR; COLORIMETRY; COMPUTER ASSISTED DIAGNOSIS; EVALUATION; FOURIER ANALYSIS; IMAGE ENHANCEMENT; INFORMATION RETRIEVAL; METHODOLOGY; SIGNAL PROCESSING;

ALGORITHMS; COLOR; COLORIMETRY; FOURIER ANALYSIS; IMAGE ENHANCEMENT; IMAGE INTERPRETATION, COMPUTER-ASSISTED; INFORMATION STORAGE AND RETRIEVAL; SIGNAL PROCESSING, COMPUTER-ASSISTED;

EID: 33847695369     PISSN: 10577149     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIP.2006.884955     Document Type: Article

References (32)

1. W. R. Hamilton, Elements of Quaternions. London, U.K.: Longmans Green, 1866.
2. T. A. Ell, "Hypercomplex Spectral Transforms," Ph.D. dissertation, Univ. Minnesota, Minneapolis, 1992.
3. T. A. Ell, "Quaternion-Fourier transforms for analysis of two-dimensional linear time-invariant partial-differential systems," in Proc. 32nd IEEE Conf. Decision and Control, San Antonio, TX, Dec. 15-17, 1993, vol. 1-4, pp. 1830-1841.
4. S. J. Sangwine, "Fourier transforms of colour images using quaternion, or hypercomplex, numbers," Electron. Lett., vol. 32, no. 21, pp. 1979-1980, Oct. 1996.
5. 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.
6. T. Bülow, "Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images," Ph.D. dissertation, Christian-Albrechts Univ., Kiel, Germany, 1999.
7. T. Bülow and G. Sommer, "Multi-dimensional signal processing using an algebraically extended signal representation," in Proc. Int. Workshop Algebraic Frames for the Perception-Action Cycle, G. Sommer and J. J. Koenderink, Eds., 1997, vol. LNCS 1315, pp. 148-163.
8. T. Bülow and G. Sommer, "Algebraically extended representations of multidimensional signals," in Proc. 10th Scandinavian Conf. Image Analysis, Lappeenrata, Finland, 1997, pp. 559-566.
9. S. J. Sangwine and T. A. Ell, "The discrete Fourier transform of a colour image," in Proc. Image Processing II Mathematical Methods, Algorithms and Applications, J. M. Blackledge and M. J. Turner, Eds., Chichester, U.K., 2000, pp. 430-441.
10. S. -C. Pei, J. -J. Ding, and J. -H. Chang, "Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT," IEEE Trans. Signal Process., vol.49, no. 11, pp. 2783-2797, Nov. 2001.
11. S. J. Sangwine, "Colour image edge detector based on quaternion convolution," Electron. Lett., vol. 34, no. 10, pp. 969-971, May 1998.
12. S. J. Sangwine and T. A. Ell, "Colour image filters based on hypercomplex convolution," Proc. Inst. Elect. Eng. - Vision, Image, and Signal Processing, vol. 147, no. 2, pp. 89-93, Apr. 2000.
13. S. J. Sangwine, B. N. Gatsheni, and T. A. Ell, "Linear colour-dependent image filtering based on vector decomposition," in Proc. EUSIPCO XI Eur. Signal Processing Conf., Toulouse, France, Sep. 3-6, 2002, vol. 11, pp. 274-277.
14. S. J. Sangwine, T. A. Ell, and C. E. Moxey, "Vector phase correlation," Electron. Lett., vol. 37, no. 25, pp. 1513-5, Dec. 2001.
15. C. E. Moxey, S. J. Sangwine, and T. A. Ell, "Hypercomplex correlation techniques for vector images," IEEE Trans. Signal Process., vol. 51, no. 7, pp. 1941-1953, Jul. 2003.
16. C. E. Moxey, S. J. Sangwine, and T. A. Ell, "Color grayscale image registration using hypercomplex phase-correlation," in Proc. IEEE Int. Conf. Image Processing, Rochester, NY, Sep. 22-25, 2002, vol. II, pp. 385-388.
17. 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 and Systems, Atlanta GA, May 12-15, 1996, vol. 2, pp. 684-687.
18. S. -C. Pei and C. -M. Cheng, "A novel block truncation coding of color images using a quaternion-moment-preserving principle," IEEE Trans. Commun., vol. 45, no. 4, pp. 583-595, Apr. 1997.
19. S. J. Sangwine and T. A. Ell, "Hypercomplex auto- and cross- correlation of color images," in Proc. IEEE Int. Conf. Image Processing, Kobe, Japan, Oct. 24-28, 1999, pp. 319-322.
20. T. A. Ell and S. J. Sangwine, "Hypercomplex Wiener-Khintchine theorem with application to color image correlation," in Proc. IEEE Int. Conf. Image Processing, Vancouver, BC, Canada, Sep. 11-14, 2000, vol. II, pp. 792-795.
21. C. E. Moxey, T. A. Ell, and S. J. Sangwine, "Vector correlation of color images," in Proc. 1st Eur. Conf. Color in Graphics, Imaging, and Vision, Apr. 2-5, 2002, pp. 343-7.
22. C. E. Moxey, T. A. Ell, and S. J. Sangwine, "Hypercomplex operators and vector correlation," in Proc. EUSIPCO XI European Signal Processing Conf., Toulouse, France, Sep. 3-6, 2002, vol. III, pp. 247-250.
23. M. Frigo and S. G. Johnson, FFTW Software Library, Mass. Inst. Technol., Cambridge, 1997 [Online]. Available: http://www.fftw.org
24. T. A. Ell and S. J. Sangwine, M. Gabbouj and P. Kuosmanen, Eds., "Decomposition of 2D hypercomplex Fourier transforms into pairs of complex Fourier transforms," in Proc. EUSIPCO 10th Eur. Signal Processing Conf., Tampere, Finland, Sep. 5-8, 2000, vol. II, pp. 1061-1064.
25. S. J. Sangwine and T. A. Ell, "Hypercomplex Fourier transforms of color images," in Proc. IEEE Int. Conf. Image Processing, Thessaloniki, Greece, Oct. 7-10, 2001, vol. I, pp. 137-140.
26. A. McCabe, G. West, and T. Caelli, "Filter techniques for complex spatio-chromatic image processing," in Proc. IEEE Int. Conf. Image Processing, Chicago, IL, Oct. 4-7, 1998, vol. 2, pp. 742-746.
27. A. D. McCabe, "A Theory of Spatio-Chromatic Image Encoding," Ph.D. dissertation, School Comput., Curtin Univ. Technol., Curtin, Australia, 1999.
28. A. McCabe, T. Caelli, G. West, and A. Reeves, "Theory of spatiochromatic image encoding and feature extraction," J. Opt. Soc. Amer., vol. 17, no. 10, pp. 1744-1754, Oct. 2000.
29. H. Palus, "Representations of colour images in different colour spaces," in The Colour Image Processing Handbook, S. J. Sangwine and R. E. N. Horne, Eds. London, U.K.: Chapman and Hall, 1998, ch. 4, pp. 67-90.
30. S. Sangwine and N. Le Bihan, Quaternion Toolbox for Matlab, Sofware Library [Online]. Available: http://qtfm.sourceforge.net/
31. C. J. Evans, S. J. Sangwine, and T. A. Ell, "Colour-Sensitive edge detection using hypercomplex filters," in Proc. EUSIPCO 10th Eur. Signal Processing Conf., M. Gabbouj and P. Kuosmanen, Eds., Tampere, Finland, Sep. 5-8, 2000, vol. I, pp. 107-110.
32. J. P. Ward, Quaternions and Cayley Numbers, Algebra and Applications. Norwell, MA: Kluwer, 1997.