Eigenstructure and fractionalization of the quaternion discrete Fourier transform

Quaternion signal processing has found significant application to color image processing and bivariate signal analysis. Among the studies on quaternion transforms, just a few deal with fractionalization and none does so from an eigenstructure analysis point of view. This work aims to contribute by defining a fractional quaternion discrete Fourier transform (with one or multiple parameters) out of its eigendecomposition, after proving that it shares eigenvectors with the usual unitary discrete Fourier transform. For illustrative purposes, an encryption scheme for color images with opacity layer is provided and discussed.