Generalized sampling expansion for the quaternion linear canonical transform

Abstract

The theory of quaternions has gained firm ground in recent times and is being widely explored, with the field of signal and image processing being no exception. However, many important aspects of quaternionic signals are yet to be explored, particularly the formulation of Generalized Sampling Expansions (GSE). In the present article, our aim is to formulate the GSE in the realm of a one-dimensional quaternion linear canonical transform (QLCT). To facilitate the intent, we construct a set of quaternion-valued filter functions which are used to construct a system of equations determining the synthesis functions for the process of reconstruction. Besides, as a special case, another sampling formula involving the derivatives of the quaternionic signal is also obtained in the sequel. Since derivatives contain information about the edges and curves appearing in images, therefore, by invoking the higher degrees of freedom pertaining to QLCT, the obtained sampling formula can be used to develop new image scaling algorithms with state-of-the-art flexibility. As an endorsement of the obtained results, an example with simulations demonstrating the signal reconstruction is presented at the end.