- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 162769, 10 pages
Convolution Theorems for Quaternion Fourier Transform: Properties and Applications
1Department of Mathematics, Hasanuddin University, Makassar 90245, Indonesia
2Division of Mathematical Sciences, Osaka Kyoiku University, Osaka 582-8582, Japan
3Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada K1N 6N5
Received 1 June 2013; Revised 1 September 2013; Accepted 7 September 2013
Academic Editor: Narcisa C. Apreutesei
Copyright © 2013 Mawardi Bahri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- J. Ebling and G. Scheuermann, “Clifford Fourier transform on vector fields,” IEEE Transactionson Visualization and Computer Graphics, vol. 11, no. 4, pp. 469–479, 2005.
- E. Hitzer and B. Mawardi, “Clifford Fourier transform on multivector fields and uncertainty principles for dimensions and ,” Advances in Applied Clifford Algebras, vol. 18, no. 3-4, pp. 715–736, 2008.
- B. Mawardi and E. Hitzer, “Clifford Fourier transformation and uncertainty principle for the Clifford geometric algebra ,” Advances in Applied Clifford Algebras, vol. 16, no. 1, pp. 41–61, 2006.
- H. de Bie and N. de Schepper, “Fractional Fourier transform of hypercomplex signals,” Signal, Image and Video Processing, vol. 6, no. 3, pp. 381–388, 2012.
- X. Guanlei, W. Xiaotong, and X. Xiagang, “Fractional quaternion Fourier transform, convolutionand correlation,” Signal Processing, vol. 88, no. 10, pp. 2511–2517, 2008.
- M. Bahri, E. Hitzer, R. Ashino, and R. Vaillancourt, “Windowed Fourier transform of two-dimensional quaternionic signals,” Applied Mathematics and Computation, vol. 216, no. 8, pp. 2366–2379, 2010.
- M. Bahri, R. Ashino, and R. Vaillancourt, “Two-dimensional quaternion wavelet transform,” Applied Mathematics and Computation, vol. 218, no. 1, pp. 10–21, 2011.
- M. Bahri, “Quaternion algebra-valued wavelet transform,” Applied Mathematical Sciences, vol. 5, no. 71, pp. 3531–3540, 2011.
- S. J. Sangwine, “Biquaternion (complexified quaternion) roots of ,” Advances in Applied Clifford Algebras, vol. 16, no. 1, pp. 63–68, 2006.
- T. Bülow, Hypercomplex spectral signal representations for the processing and analysis of images [Ph.D. thesis], University of Kiel, Kiel, Germany, 1999.
- T. A. Ell, “Quaternion-Fourier transformations for analysis of two-dimensional lineartime-invariant partial differential systems,” in Proceedings of the 32nd IEEE Conference on Decision and Control, vol. 2, pp. 1830–1841, San Antonio, Tex, USA, December 1993.
- E. Hitzer, “Quaternion Fourier transform on quaternion fields and generalizations,” Advances in Applied Clifford Algebras, vol. 17, no. 3, pp. 497–517, 2007.
- M. Bahri, E. Hitzer, A. Hayashi, and R. Ashino, “An uncertainty principle for quaternion Fourier transform,” Computers & Mathematics with Applications, vol. 56, no. 9, pp. 2411–2417, 2008.
- D. Assefa, L. Mansinha, K. F. Tiampo, H. Rasmussen, and K. Abdella, “Local quaternion Fourier transform and color images texture analysis,” Signal Processing, vol. 90, no. 6, pp. 1825–1835, 2010.
- R. Bujack, G. Scheuermann, and E. Hitzer, “A general geometric fourier transform convolution theorem,” Advances in Applied Clifford Algebras, vol. 23, no. 1, pp. 15–38, 2013.
- R. Bujack, G. Scheuermann, and E. Hitzer, “A general geometric Fourier transform,” in Proceedings of the 9th International Conference on Clifford Algebras and Their Applications in Mathematical Physics (ICCA '11), K. Guerlebeck, Ed., Weimar, Germany, July 2011.
- E. Hitzer, “Directional uncertainty principle for quaternion Fourier transform,” Advances in Applied Clifford Algebras, vol. 20, no. 2, pp. 271–284, 2010.
- G. B. Folland, Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 2nd edition, 1999.
- T. A. Ell and S. J. Sangwine, “Hypercomplex Fourier transforms of color images,” IEEE Transactions on Image Processing, vol. 16, no. 1, pp. 22–35, 2007.
- S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego, Calif, USA, 2nd edition, 1999.
- M. Petrou and C. Petrou, Image Processing: The Fundamentals, John Wiley & Sons, West Sussex, UK, 2nd edition, 2010.
- L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Springer, Berlin, Germany, 1983.