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Connecting spatial and frequency domains for the quaternion Fourier transform

De Bie, H and De Schepper, N and Ell, TA and Rubrecht, K and Sangwine, SJ (2015) 'Connecting spatial and frequency domains for the quaternion Fourier transform.' Applied Mathematics and Computation, 271. 581 - 593. ISSN 0096-3003

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Abstract

The quaternion Fourier transform (qFT) is an important tool in multi-dimensional data analysis, in particular for the study of color images. An important problem when applying the qFT is the mismatch between the spatial and frequency domains: the convolution of two quaternion signals does not map to the pointwise product of their qFT images. The recently defined ‘Mustard’ convolution behaves nicely in the frequency domain, but complicates the corresponding spatial domain analysis. The present paper analyses in detail the correspondence between classical convolution and the new Mustard convolution. In particular, an expression is derived that allows one to write classical convolution as a finite linear combination of suitable Mustard convolutions. This result is expected to play a major role in the further development of quaternion image processing, as it yields a formula for the qFT spectrum of the classical convolution.

Item Type: Article
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Divisions: Faculty of Science and Health > Computer Science and Electronic Engineering, School of
Depositing User: Jim Jamieson
Date Deposited: 09 Oct 2015 13:28
Last Modified: 06 Oct 2020 15:15
URI: http://repository.essex.ac.uk/id/eprint/15236

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