Abstract
In this paper we use the general two-sided quaternion Fourier transform (QFT), and relate the classical convolution of quaternion-valued signals over \({{\mathbb R}^2}\) with the Mustard convolution. A Mustard convolution can be expressed in the spectral domain as the point wise product of the QFTs of the factor functions. In full generality do we express the classical convolution of quaternion signals in terms of finite linear combinations of Mustard convolutions, and vice versa the Mustard convolution of quaternion signals in terms of finite linear combinations of classical convolutions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bas, P.; Le Bihan, N.; Chassery, J.M.: Color image watermarking using quaternion Fourier transform. In: Acoustics, Speech, and Signal Processing, 2003. Proceedings (ICASSP03). IEEE International Conference on vol. 3, pp. III-521 (2003)
Bayro-Corrochano E.: The theory and use of the quaternion wavelet transform. J. Math. Imaging Vis. 24, 19–35 (2006)
Bayro-Corrochano E., Trujillo N., Naranjo M.: Quaternion Fourier descriptors for the preprocessing and recognition of spoken words using images of spatiotemporal representations. J. Math. Imaging Vis. 28, 179–190 (2007)
Bülow, T.: Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images. PhD Thesis, Univ. of Kiel (1999)
Bujack R., De Bie H., De Schepper N., Scheuermann G.: Convolution products for hypercomplex Fourier transforms. J. Math. Imaging Vis. 48, 606–624 preprint: http://arxiv.org/abs/1303.1752 (2014)
Coxeter H.S.M.: Quaternions and reflections. Am. Math. Mon. 53(3), 136–146 (1946)
De Bie H., De Schepper N., Ell T.A., Rubrecht K., Sangwine S.J.: Connecting spatial and frequency domains for the quaternion Fourier transform. Appl. Math. Comput. 271, 581–593 (2015)
Denis P., Carré P., Fernandez-Maloigne C.: Spatial and spectral quaternionic approaches for colour images. Comput. Vis. Image Underst. 107, 74–87 (2007)
Ell, TA.: Quaternionic-Fourier transform for analysis of two-dimensional linear time-invariant partial differential systems. In: Proceedings of the 32nd IEEE Conference on Decision and Control, December 15–17, vol. 2, pp. 1830–1841 (1993)
Ell, TA., Sangwine, SJ.: Hypercomplex Fourier transforms of color images. IEEE Trans. Image Process. 16(1), 22–35 (2007)
Guo C., Zhang L.: A novel multiresolution spatiotemporal saliency detection model and its applications in image and video compression. IEEE Trans. Image Process. 19, 185–198 (2010)
Hildenbrand, D.: Foundations of Geometric Algebra Computing, Springer, Berlin (2013)
Hitzer, E.: Quaternion Fourier transform on quaternion fields and generalizations. Adv. Appl. Clifford Algebra 17, 497–517 (2007). doi:10.1007/s00006-007-0037-8, preprint: http://arxiv.org/abs/1306.1023
Hitzer, E.: Directional uncertainty principle for quaternion Fourier transforms, Adv. Appl. Clifford Algebra 20(2), 271–284 (2010). doi:10.1007/s00006-009-0175-2, preprint: http://arxiv.org/abs/1306.1276
Hitzer, E.: OPS-QFTs: A new type of quaternion Fourier transforms based on the orthogonal planes split with one or two general pure quaternions. In: Numerical Analysis and Applied Mathematics ICNAAM 2011, AIP Conf. Proc. 1389, pp. 280–283 (2011). doi:10.1063/1.3636721, preprint: http://arxiv.org/abs/1306.1650
Hitzer, E.: Creative Peace License. http://gaupdate.wordpress.com/2011/12/14/the-creative-peace-license-14-dec-2011/
Hitzer, E.; Sangwine, SJ.: The orthogonal 2D planes split of quaternions and steerable quaternion Fourier transformations, in: E. Hitzer and S.J. Sangwine (Eds.), Quaternion and Clifford Fourier Transforms and Wavelets, Trends in Mathematics, vol. 27, pp. 15–40. Birkhäuser (2013) doi:10.1007/978-3-0348-0603-9_2, preprint: http://arxiv.org/abs/1306.2157
Hitzer, E., Helmstetter, J., Abłamowicz, R.: Square Roots of 1 in Real Clifford Algebras. In: Hitzer, E., Sangwine S.J. (eds.) Quaternion and Clifford Fourier Transforms and Wavelets, Trends in Mathematics, vol. 27, pp. 123–153. Birkhäuser (2013). doi:10.1007/978-3-0348-0603-9_7, preprint: http://arxiv.org/abs/1204.4576
Hitzer, E.: Two-Sided Clifford Fourier Transform with Two Square Roots of 1 in Cl(p, q). Adv. Appl. Clifford Algebras 24, 313–332 (2014). doi:10.1007/s00006-014-0441-9, preprint: http://arxiv.org/abs/1306.2092
Hitzer, E.: The orthogonal planes split of quaternions and its relation to quaternion geometry of rotations. IOP J. Phys. Conf. Ser. (JPCS), 597, 012042 (2015). doi:10.1088/1742-6596/597/1/012042. Open Access URL: http://iopscience.iop.org/1742-6596/597/1/012042.pdf, preprint: http://arXiv.org/abs/1411.0362
Jin L., Liu H., Xu X., Song E.: Quaternion-based impulse noise removal from color video sequences. IEEE Trans. Circ. Syst. Video Technol. 23, 741–755 (2013)
Meister L., Schaeben H.: A concise quaternion geometry of rotations. Math. Meth. Appl. Sci. 28, 101–126 (2005)
Moxey C.E., Sangwine S.J., Ell T.A.: Hypercomplex correlation techniques for vector images. IEEE Trans. Signal Process. 51, 1941–1953 (2003)
Mustard D.: Fractional convolution. J. Aust. Math. Soc. Ser. B Vol. 40, 257–265 (1998)
Sangwine S.J.: Color image edge detector based on quaternion convolution. Electron. Lett. 34, 969–971 (1998)
Sangwine S.J.: Fourier transforms of colour images using quaternion, or hypercomplex, numbers. Electron. Lett. 32(21), 1979–1980 (1996)
Sangwine, S.J., Le Bihan, N.: Quaternion and octonion toolbox for Matla. http://qtfm.sourceforge.net/. Accessed 29 Mar 2016
Soulard R., Carré P.: Quaternionic wavelets for texture classification. Pattern Recognit. Lett. 32, 1669–1678 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Soli Deo Gloria.
In memory of Mrs. Lucy Baker, †31 December 2015, who worked with Seeds Of Hope Foundation, Mumbai, India. The use of this paper is subject to the Creative Peace License [16].
Rights and permissions
About this article
Cite this article
Hitzer, E. General two-sided quaternion Fourier transform, convolution and Mustard convolution. Adv. Appl. Clifford Algebras 27, 381–395 (2017). https://doi.org/10.1007/s00006-016-0684-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00006-016-0684-8