The Continuous quaternion Algebra-Valued Wavelet Transform and the Associated Uncertainty Principle

2019/02/22 by Youssef El Haoui, Haoui, Youssef El, Saïd Fahlaoui +1
Computer Science · Engineering · Mathematics · #11R52 #30G35 #42B10 #42C40 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #Sparse and Compressive Sensing Techniques

paper · pdf · doi:10.48550/arxiv.1902.08461

openalex publication_date 2019/02/22 · openalex created_date 2019/12/26 · openalex updated_date 2026/07/28

Abstract

The purpose of this article is to extend the wavelet transform to quaternion algebra using the kernel of the two-sided quaternion Fourier transform (QFT). We study some fundamental properties of this extension such as scaling, translation, rotation, Parseval's identity, inversion theorem, and a reproducing kernel, then we derive the associated Heisenberg-Pauli-Weyl uncertainty principle UP. Finally, using the quaternion Fourier representation of the CQWT we generalize the logarithmic UP and Hardy's UP to the CQWT domain.

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