Connecting spatial and frequency domains for the quaternion Fourier transformConnecting spatial and frequency domains for the quaternion Fourier transform
Faculty of Sciences. Mathematics and Computer Science
2015New York, N.Y., 2015
Applied mathematics and computation. - New York, N.Y.
271(2015), p. 581-593
University of Antwerp
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. (C) 2015 Elsevier Inc. All rights reserved.