International Journal of Image, Graphics and Signal Processing(IJIGSP)

ISSN: 2074-9074 (Print), ISSN: 2074-9082 (Online)

Published By: MECS Press

IJIGSP Vol.4, No.2, Mar. 2012

Applying Quaternion Fourier Transforms for Enhancing Color Images

Full Text (PDF, 922KB), PP.9-15

Views:152   Downloads:4


M.I. Khalil

Index Terms

Image processing,Fourier transforms,Hypercomplex 2D Fourier transform, quaternions,Bandpass Filter


The Fourier transforms play a critical role in a broad range of image processing applications, including enhancement, analysis, restoration, and compression. Until recently, it was common to use the conventional methods to deal with colored images. These methods are based on RGB decomposition of the colored image by separating it into three separate scalar images and computing the Fourier transforms of these images separately. The computing of the Hypercomplex 2D Fourier transform of a color image as a whole unit has only recently been realized. This paper is concerned with frequency domain noise reduction of color images using quaternion Fourier transforms. The approach is based on obtaining quaternion Fourier transform of the color image and applying the Gaussian filter to it in the frequency domain. The filtered image is then obtained by calculating the inverse quaternion Fourier transforms.

Cite This Paper

M.I. Khalil,"Applying Quaternion Fourier Transforms for Enhancing Color Images", IJIGSP, vol.4, no.2, pp.9-15, 2012.


[1]Cooley, James W., and John W. Tukey, 1965", An algorithm for the machine calculation of complex Fourier series", Math. Comput, Vol.19, 1965, pp. 297-301.

[2]Sangwine, S., Ell, T.A., "Hypercomplex Fourier Transforms of Color Images", IEEE International Conference on Image Processing (ICIP), 2001, Vol. 1, pp. 137–140.

[3]Pei, S., Cheng, C., "A novel block truncation coding of color images by using quaiemion-moment preserving principle", IEEE International Symposium on Circuits and systems, 1996, Vol. 2, pp. 684–687.

[4]Sangwine, S., Ell, T, "Hypercomplex fourier transforms of color images", IEEE International Conference on Image Processing, 2001, Vol.1, pp. 137–140.

[5]Bihan, N.L., Sangwine, S.J., "Quaternion principal component analysis of color images", IEEE International Conference on Image Processing, 2003, Vol, 1, pp. 809–812.

[6]Ja-Han Chang, S.C.P., Ding, J.J., "2d quaternion fourier spectral analysis and its applications", IEEE International Symposium on Circuits and Systems, 2004, Vol. 3, pp. 241–244.

[7]S. J. Sangwine, "Fourier transforms of colour images using quaternion, or hypercomplex, numbers", Electron. Lett. Vol. 32, No. 21, 1996, pp.1979–1980..

[8]Ell T.A., "Quaternion-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems," in Proc. 32nd Con. Decision Contr., Dec. 1993, pp. 1830-1841.

[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]Sangwine S., "Colour image edge detector based on quaternion convolution", Electronic Lett, Vol. 34, No. 10, 1998; pp. 969–971.

[11]Moxey C, Sangwine S, "Ell T. Hypercomplex correlation techniques for vector images", IEEE Transactions on Signal Processing, Vol. 51, No.7, 1941.

[12]Subakan ÖN, Vemuri BC, "Image segmentation via convolution of a level-set function with a Rigaut kernel",IEEE Conference on Computer Vision and Pattern Recognition; Anchorage, Alaska. June 2008.

[13]Chan TF, Yezrielev B, Vese LA., "Active contours without edges for vector-valued", images. Journal of Visual Communication and Image Representation, Vol. 11, 2000, pp. 130–141.

[14]Sangwine, S.J., "Colour in Image Processing", Electronics and Communication Engineering Journal, Vol. 12, No. 5, pp.211-219, October 2000.