Abstract
This paper deals with the construction of \(n=3 \text{ mod } 4\) Clifford algebra \(Cl_{n,0}\)-valued admissible shearlet transform using the shearlet group \((\mathbb {R}^* < imes \mathbb {R}^{n-1}) < imes \mathbb {R}^n\), a subgroup of affine group of \({\mathbb {R}}^n\). The admissibility conditions for a nonzero Clifford valued square integrable function have been obtained. Various properties such as reconstruction formula, orthogonality relation, isometry and reproducing kernel have been dealt. As an application, Heisenberg type uncertainty principle for Clifford shearlet transform has been derived.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bahri, M., Adji, S., Zhao, J.: Clifford algebra-valued wavelet transform on multivector fields. Adv. Appl. Clifford Algebr. 21(1), 13–30 (2011)
Bahri, M., Ashino, R., Vaillancourt, R.: Continuous quaternion Fourier and wavelet transforms. Int. J. Wavelets Multiresolut. Inf. Process. 12(4), 1460003 (2014). (21 pp)
Bahri, M., Hitzer, E.S.M.: Clifford algebra \({\rm Cl}_{3,0}\)-valued wavelet transformation, Clifford wavelet uncertainty inequality and Clifford Gabor wavelets. Int. J. Wavelets Multiresolut. Inf. Process. 5(6), 997–1019 (2007)
Banouh, H., Ben Mabrouk, A., Kesri, M.: Clifford wavelet transform and the uncertainty principle. Adv. Appl. Clifford Algebr. 29(5), 23 (2019). (Art. 106)
Brahim, K., Nefzi, B., Tefjeni, E.: Uncertainty principles for the continuous quaternion Shearlet transform. Adv. Appl. Clifford Algebr. 29(3), 33 (2019). (Art. 43)
Brackx, F., et al.: Generalized Hermitean Clifford–Hermite polynomials and the associated wavelet transform. Math. Methods Appl. Sci. 32(5), 606–630 (2009)
Brackx, F., De Schepper, N., Sommen, F.: The Clifford–Fourier transform. J. Fourier Anal. Appl. 11(6), 669–681 (2005)
Brackx, F., Sommen, F.: Clifford–Hermite wavelets in Euclidean space. J. Fourier Anal. Appl. 6(3), 299–310 (2000)
Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis, Research Notes in Mathematics, 76, Pitman. Advanced Publishing Program, Boston (1982)
Dahlke, S., et al.: The uncertainty principle associated with the continuous shearlet transform. Int. J. Wavelets Multiresolut. Inf. Process. 6(2), 157–181 (2008)
El Haoui, Y., Fahlaoui, S.: Donoho–Stark’s uncertainty principles in real Clifford algebras. Adv. Appl. Clifford Algebr. 29(5), 13 (2019). (Paper No. 94)
Hestenes, D.: New Foundations for Xlassical Mechanics, Fundamental Theories of Physics. D. Reidel Publishing Co., Dordrecht (1986)
Hestenes, D., Sobczyk, G.: Clifford algebra to geometric calculus, Fundamental Theories of Physics. D. Reidel Publishing Co., Dordrecht (1984)
Hitzer, E.M.S., Mawardi, B.: Clifford Fourier transform on multivector fields and uncertainty principles for dimensions \(n=2~({\rm mod} 4)\) and \(n=3~({\rm mod} 4)\). Adv. Appl. Clifford Algebr. 18(3–4), 715–736 (2008)
E. S. M. Hitzer and B. Mawardi, Uncertainty principle for Clifford geometric algebras \({\rm Cl}_{n,0}, n=3 ({\rm mod} 4)\) based on Clifford Fourier transform, in Wavelet analysis and applications, 47–56, Appl. Numer. Harmon. Anal, Birkhäuser, Basel
El Kamel, J., Jday, R.: Uncertainty principles for the Clifford–Fourier transform. Adv. Appl. Clifford Algebr. 27(3), 2429–2443 (2017)
Mawardi, B., Hitzer, E.M.S.: Clifford Fourier transformation and uncertainty principle for the Clifford geometric algebra \({{\rm Cl}}_{3,0}\). Adv. Appl. Clifford Algebr. 16(1), 41–61 (2006)
Acknowledgements
The authors would like to express their sincere gratitude to the referees for their insightful and valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Eckhard Hitzer
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sharma, J., Singh, S.K. Clifford Valued Shearlet Transform. Adv. Appl. Clifford Algebras 30, 38 (2020). https://doi.org/10.1007/s00006-020-01066-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00006-020-01066-8