Hypercomplex colour image processing

This page describes current work on hypercomplex processing of digital colour images in the Multimedia Architectures and Applications Research Group in the Department of Electronic Systems Engineering at the University of Essex. The research described here was started at The University of Reading and some details of related work completed at Reading are also available.

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.

[Lena image] [Autocorrelation modulus] [Autocorrelation a] [Autocorrelation b]

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:

Publications

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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.
  8. 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.
  9. 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.]
  10. Sangwine, S. J., 'Fourier transforms of colour images using quaternion, or hypercomplex, numbers', Electronics Letters, 32, (21), October 10 1996, 1979-80.

Hypercomplex Filtering of Digital Colour Images

This project is supported by EPSRC Grant GR/M45764, and started in July 1999. Dr Carolyn J. Evans (now at CSIRO in Sydney) was the first Research Fellow to work on this project (for about ten months, at the University of Reading). From March 2001 until May 2002, Dr Eddie Moxey worked on the project, initially part-time, but subsequently full-time. From September 2001, Dr Barnabas Gatsheni has been a second Research Officer on the project. The objective of the work is to develop new colour image filters based on convolution using hypercomplex masks, and to understand the theory of hypercomplex filtering. We have developed two new colour edge detecting filters, published in conference papers, and we have also studied vector correlation. We presented a poster at a one-day meeting held in London in January 2000 (available below).

Publications

  1. 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.
  2. Sangwine, S. J., Ell, T. A. and Moxey, C. E., 'Vector Phase Correlation', Electronics Letters, 37, (25), December 6 2001, 1513-5.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. Moxey, C. E., Sangwine, S. J. and Ell, T. A., 'Color-grayscale image registration using hypercomplex phase-correlation', ibid., II, 385-388.
  8. 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.
  9. Sangwine, S. J., Gatsheni, B. N. and Ell, T. A., 'Linear colour-dependent image filtering based on vector decomposition', ibid., II, 274-277.

Hypercomplex signal processing

Since 2001 a collaboration has developed with Dr Nicolas le Bihan at LIS, INPG, Grenoble. This collaboration is now supported by an EPSRC Visiting Fellowship (Grant GR/S16881/01) valid for 2 years. The aim of the work is to extend hypercomplex matrix methods in signal processing and image processing, including the singular value decomposition. Several conference papers are already accepted and will be listed here shortly.
Compiled and edited by Dr Steve Sangwine
Last updated: 14 March 2007

To: Department of Electronic Systems Engineering

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