Tensor transform-based quaternion fourier transform algorithm

Abstract

The traditional Fourier transform is a fundamental tool in signal and image processing, which in color imaging transforms images either in monochrome or as separate color channels, thereby ignoring interactions between the color channels. The quaternion discrete Fourier transform (QDFT), a generalization of the Fourier transform, represents the three color channels of an image in the RGB color space as a vector field in the quaternion arithmetic allowing for simultaneous analysis of all color data. It is expected that this new color imaging model which captures the interactions between the different color channels will improve upon existing image processing and recognition systems. Efficient QDFT algorithms are derived by combining classical DFT transforms, enabling standard fast Fourier transform (FFT) software to provide a fast QDFT numerical implementation. We developed an open-source quaternion Fourier transform tool that calculates the 2-D QDFT by the column-row method which is performed by calculating multiple 1-D QDFTs where each 1-D transform is calculated by two complex 1-D DFTs. The matrices involved in these calculations necessitate the inclusion of a discussion on symplectic decomposition. Our main contribution is the presentation a new algorithm for the tensor representation of the QDFT that is structurally simpler and uses less multiplications than its predecessors. For instance, for the images of size N×N, where N=2r,r>2, the tensor 2-D QDFT saves about 2N2r real multiplications.