The Quaternion-Fourier Transform and Applications

It is well-known that the Fourier transforms plays a critical role in image processing and the corresponding applications, such as enhancement, restoration and compression. For filtering of gray scale images, the Fourier transform in \(\mathbb {R}^2\) is an important tool which converts the image from spatial domain to frequency domain, then by applying filtering mask filtering is done. To filter color images, a new approach is implemented recently which uses hypercomplex numbers (called Quaternions) to represent color images and uses Quaternion-Fourier transform for filtering. The quaternion Fourier transform has been widely employed in the colour image processing. The use of quaternions allow the analysis of color images as vector fields, rather than as color separated components. In this paper we mainly focus on the theoretical part of the Quaternion Fourier transform: the real Paley-Wiener theorems for the Quaternion-Fourier transform on \(\mathbb {R}^2\) for Quaternion-valued Schwartz functions and \(L^p\)-functions, which generalizes the recent results of real Paley-Wiener theorems for scalar- and quaternion-valued \(L^2\)-functions.