Research on hypercomplex colour image processing started in 1996 and
arose from the work of Dr Amy Thornton
who applied complex Fourier transforms to chrominance images
(see here).
The quaternions are a generalization of the complex numbers
with an imaginary part consisting of three components, and the idea
was to use these to generalize techniques from signal and image processing,
such as frequency domain filtering and correlation, to handle colour
images *as vector images*.
More details about quaternions and links to general information elsewhere are
given on a separate page.

The first development, in 1996, was a quaternion Fast Fourier Transform (FFT) based on the work of Dr Todd A. Ell at the University of Minnesota and published in 1993. Subsequently a collaboration has developed and recently (June 2003) this collaboration has been supported by the award of an EPSRC Visiting Fellowship (Grant no GR/S58621/01) valid for 3 years. In 1998, a new colour edge detector was developed using quaternion convolution. The result of applying this edge detector to the Lena image is shown on a separate page.

A significant development in the quaternion work is the definition of auto-correlation and cross-correlation of colour images using quaternion correlation. The following images show the results.

This work was presented in poster form at the IEEE International Conference on Image Processing (ICIP99) in Kobe, Japan, in October 1999. The full paper is available in the conference proceedings (IEEE copyright), and the poster is available here:

- HYPERCOMPLEX AUTO- AND CROSS-CORRELATION OF COLOR IMAGES

Stephen J. Sangwine and Todd A. Ell

A0 format (1188 by 839 mm) PDF (320 kB)

- Sangwine, S. J. and Ell, T. A., 'Mathematical approaches to linear vector filtering of color images', First European Conference on Colour in Graphics, Imaging and Vision (CGIV 2002), University of Poitiers, France, 2-5 April 2002, The Society for Imaging Science and Technology, 348-351.
- Sangwine, S. J. and Ell, T. A., 'Hypercomplex Fourier Transforms of Color Images',
IEEE International Conference on Image Processing (ICIP 2001), Thessaloniki, Greece,
October 7-10, 2001,
**I**, 137-140. - Sangwine, S. J. and Ell, T. A.,
'Colour image filters based on hypercomplex convolution',
*IEE Proceedings - Vision, Image and Signal Processing*,**147**, (2), April 2000, 89-93. - Ell, T. A. and Sangwine, S. J., 'Hypercomplex Wiener-Khintchine Theorem with Application
to Color Image Correlation', IEEE International Conference on Image Processing (ICIP 2000),
Vancouver, Canada, September 11-14, 2000,
**II**, 792-795. - Ell, T. A. and Sangwine, S. J., 'Decomposition of 2D Hypercomplex Fourier Transforms into
Pairs of Complex Fourier Transforms', in: Gabbouj, M. and Kuosmanen (eds),
'Signal Processing X, Theories and Applications', Proceedings of EUSIPCO 2000,
Tenth European Signal Processing Conference, Tampere, Finland,
5-8 September 2000,
**II**, 1061-1064. - Sangwine, S. J. and Ell, T. A., 'The discrete Fourier transform of a colour image', in Blackledge, J. M. and Turner, M. J. (eds), 'Image Processing II: Mathematical Methods, Algorithms and Applications', (Proceedings of Second IMA Conference on Image Processing, De Montfort University, Leicester, UK, September 1998), Horwood Publishing for Institute of Mathematics and its Applications, 2000, 430-441.
- Sangwine, S. J. and Ell, T. A., 'Hypercomplex auto- and cross-correlation of color images',
IEEE International Conference on Image Processing, (ICIP'99), Kobe, Japan, 24-28 October 1999,
**IV**, 319-322, also on CD-ROM ISBN 0-7803-5470-2. - Sangwine, S. J., 'The problem of defining the Fourier transform of a colour image',
IEEE International Conference on Image Processing, (ICIP'98), Chicago, USA, October 4-7 1998,
**I**, 171-175. - Sangwine, S. J.,
'Colour image edge detector based on quaternion convolution',
*Electronics Letters*,**34**, (10), May 14 1998, 969-971. [This paper contains colour images which are printed in monochrome in the paper edition of the journal. The colour images are available here.] - Sangwine, S. J.,
'Fourier transforms of colour images using quaternion,
or hypercomplex, numbers',
*Electronics Letters*,**32**, (21), October 10 1996, 1979-80.

- PROCESSING OF COLOUR IMAGES USING HYPERCOMPLEX NUMBERS

Stephen J. Sangwine, Carolyn J. Evans and Todd A. Ell

A0 format (1188 by 839 mm) PDF (1147 kB).

- Moxey, C. E., Sangwine, S. J. and Ell, T. A. 'Hypercomplex Correlation Techniques for Vector Images',
*IEEE Transactions on Signal Processing*,**51,**, (7), July 2003, 1941-1953. - Sangwine, S. J., Ell, T. A. and Moxey, C. E., 'Vector Phase Correlation',
*Electronics Letters*,**37**, (25), December 6 2001, 1513-5. - Moxey, C. E., Ell, T. A. and Sangwine, S. J., 'Vector correlation of color images', First European Conference on Colour in Graphics, Imaging and Vision (CGIV 2002), University of Poitiers, France, 2-5 April 2002, The Society for Imaging Science and Technology, 343-7.
- Evans, C.J., Ell, T. A. and Sangwine, S. J.,
'Hypercomplex Color-Sensitive Smoothing Filters',
IEEE International Conference on Image Processing (ICIP 2000), Vancouver, Canada,
September 11-14, 2000,
**I**, 541-544. - Evans, C.J., Sangwine, S. J. and Ell, T. A.,
'Colour-Sensitive Edge Detection Using Hypercomplex Filters',
in: Gabbouj, M. and Kuosmanen (eds), 'Signal Processing X, Theories and Applications',
Proceedings of EUSIPCO 2000, Tenth European Signal Processing Conference, Tampere, Finland,
5-8 September 2000,
**I**, 107-110. - Sangwine, S. J. and Ell, T. A., 'Vector zone plates as test patterns for linear vector filters',
IEEE International Conference on Image Processing (ICIP 2002), Rochester, NY, USA, 22-25 September
2002,
**II**, 361-364. - Moxey, C. E., Sangwine, S. J. and Ell, T. A., 'Color-grayscale image registration using hypercomplex
phase-correlation',
*ibid.*,**II**, 385-388. - Moxey, C. E., Ell, T. A. and Sangwine, S. J., 'Hypercomplex operators and vector correlation',
EUSIPCO 2002, Eleventh European Signal Processing Conference, 3-6 September 2002, Toulouse,
France,
**III**, 247-250. - Sangwine, S. J., Gatsheni, B. N. and Ell, T. A., 'Linear colour-dependent image filtering based on
vector decomposition',
*ibid.*,**II**, 274-277.

Last updated: 14 March 2007