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Discrete Wavelets with Quaternion and Clifford Coefficients

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Abstract

In this paper we use matrix representations of quaternions and Clifford algebras and solve the same matrix equations in each case to find Daubechies quaternion and Clifford scaling filters. We use paraunitary completion of the polyphase matrix to find corresponding quaternion and Clifford wavelet filters. We then use the cascade algorithm on our filters to find quaternion and Clifford scaling and wavelet functions, which we illustrate using all possible projections onto two and three dimensions: to our knowledge, this is the first time that this has been done. We discuss the shapes of these functions and conclude with a consideration of what we could actually do with our filters.

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Acknowledgements

I should like to thank my PhD supervisor, Steve Sangwine, for his help with the writing of the later parts of this article.

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  1. School of Computer Science and Electronic Engineering, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, UK

    Peter Fletcher

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  1. Peter Fletcher
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Correspondence to Peter Fletcher.

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This article is part of the Topical Collection on Proceedings ICCA 11, Ghent, 2017, edited by Hennie De Schepper, Fred Brackx, Joris van der Jeugt, Frank Sommen, and Hendrik De Bie.

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Fletcher, P. Discrete Wavelets with Quaternion and Clifford Coefficients. Adv. Appl. Clifford Algebras 28, 59 (2018). https://doi.org/10.1007/s00006-018-0876-5

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