Abstract
Gabor frames play a vital role not only in modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this work we present a non-trivial generalization of Gabor frames to the quaternionic case and give new density results. The key tool is the two-sided windowed quaternionic Fourier transform (WQFT). As in the complex case, we want to write the WQFT as an inner product between a quaternion-valued signal and shifts and modulates of a real-valued window function. We demonstrate a Heisenberg uncertainty principle and for the results regarding the density, we employ the quaternionic Zak transform to obtain necessary and sufficient conditions to ensure that a quaternionic Gabor system is a quaternionic Gabor frame. We conclude with a proof that the Gabor conjecture does not hold true in the quaternionic case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bahri, M., Ashino, R.: Two-dimensional quaternionic windowed Fourier transform. In: Nikolic, G. (ed.) Fourier Transforms—Approach to Scientific Principles, pp. 247–260 (2011)
Bahri M., Hitzer E., Ashino R., Vaillancourt R.: Windowed fourier transform of two-dimensional quaternionic signals. Appl. Math. Comput. 216, 2366–2379 (2010)
Bahri M., Hitzer E., Hayashi A., Ashino R.: An uncertainty principle for quaternion Fourier transform. Comput. Math. Appl. 56, 2411–2417 (2008)
Bas, P., Le Bihan, N., Chassery, J.M: Color image watermarking using quaternion Fourier transform. In: Proceedings of the IEEE International Conference on Acoustics and Signal Processing, pp. 521–524 (2003)
Borichev A., Gröchenig K., Lyubarskii Y.: Frame constants of gabor frames near the critical density. J. Math. Pures Appl. 94, 170–182 (2010)
Bülow, T.: (1999) Hypercomplex spectral signal representations for the processing and analysis of images. Ph.D. thesis, Christian-Albrechts-Universität Kiel
Ell, T.A: (2013) Quaternion Fourier transform: re-tooling image and signal processing analysis. In: Hitzer, E., Sangwine, S. (eds.) Quaternion and Clifford Fourier Transforms and Wavelets. S**er, Berlin
Ell T.A., LeBihan N., Sangwine S.J.: Quaternion Fourier Transforms for Signal and Image Processing. Wiley, New York (2014)
Fu Y., Kähler U., Cerejeiras P.: The Balian–Low theorem for the windowed quaternionic Fourier transform. Adv. Appl. Clifford Algebras 22, 1025–1040 (2012)
Gröchenig, K.: Foundations of time-frequency analysis, 2001 edn. Springer, Berlin (2001)
Gürlebeck, K., Habetha, K., Sprössig, W.: Funktionentheorie in der Ebene und im Raum, 1. aufl. edn. Springer, Berlin (2006)
Heil C.: History and evolution of the density theorem for Gabor frames. J. Four. Anal. Appl. 13(2), 113–166 (2007)
Hitzer E.: Quaternion Fourier transform on quaternion fields and generalizations. Adv. Appl. Clifford Algebras 17, 497–517 (2007)
Janssen A.J.E.M.: On generating tight gabor frames at critical density. J. Four. Ser. Appl. 9(2), 175–214 (2003)
Janssen, A.J.E.M.: Classroom proof of the density theorem for Gabor systems (ESI preprint 1649) (2005). Available at: http://www.esi.ac.at/static/esiprpr/esi1649.pdf
Seip K.: Density theorems for sampling and interpolation in the Bargmann–Fock space i. Reine Angew. Math. 429, 91–106 (1992)
Seip K., Wallsten R.: Density theorems for sampling and interpolation in the Bargmann–Fock space ii. Reine Angew. Math. 429, 107–113 (1992)
Shapiro, M., Tovar, L.M.: On a class of integral representations related to the two-dimensional helmholtz operator. In: Contemporary Mathematics 212, pp. 229–244 (1998)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hartmann, S. Some Results on the Lattice Parameters of Quaternionic Gabor Frames. Adv. Appl. Clifford Algebras 26, 137–149 (2016). https://doi.org/10.1007/s00006-015-0587-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00006-015-0587-0