New Sampling Expansion Related to Derivatives in Quaternion Fourier Transform Domain

The theory of quaternions has gained a 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 Fourier transform. We have designed quaternion Fourier filters to reconstruct the signal, using the signal and its derivative. Since derivatives contain information about the edges and curves appearing in images, therefore, such a sampling formula is of substantial importance for image processing, particularly in image super-resolution procedures. Moreover, the presented sampling expansion can be applied in the field of image enhancement, color image processing, image restoration and compression and filtering, etc. Finally, an illustrative example is presented to demonstrate the efficacy of the proposed technique with vivid simulations in MATLAB.

Automated summary

This article highlights the importance of formulating Generalized Sampling Expansions (GSE) in the realm of quaternions, especially in the context of one-dimensional quaternion Fourier transform, and demonstrates its significant role in image processing.