Discrete quaternion linear canonical transform

Fourier transform (FT), and its generalizations, the fractional Fourier transform (FrFT) and linear canonical transform (LCT) are integral transforms that are useful in optics, signal processing, and in many other fields. In the applications, the performance of LCT is superior because of its three extra degrees of freedom as compared to no degree of freedom for FT and one degree of freedom for FrFT. Recently, quaternion linear canonical transform (QLCT), an extension of the LCT in quaternion algebra, has been derived and since received noticeable attention, thanks to its elegance and expressive power in the study of multi-dimensional signals/images. To the best of our knowledge computation of the QLCT by using digital techniques is not possible now, because a discrete version of the QLCT is undefined. It initiated us to introduce the two-dimensional (2D) discrete quaternion linear canonical transform (DQLCT) that is analogous to the 2D discrete quaternion Fourier transform (DQFT). The main properties of the 2D DQLCT, including the basic properties, reconstruction formula and Rayleigh-Plancherel theorem, are obtained. Importantly, the convolution theorem and fast computation algorithm of 2D DQLCT, which are key to engineering usage, are considered. Finally, we demonstrate applications, illustrate simulations, and discuss some future prospects of the DQLCT.