Eigenfunctions, eigenvalues, and fractionalization of the quaternion and biquaternion fourier transforms

The discrete quaternion Fourier transform (DQFT) is useful for signal analysis and image processing. In this paper, we derive the eigenfunctions and eigenvalues of the DQFT. We also extend our works to the reduced biquaternion case, i.e., the discrete reduced biquaternion Fourier transform (DRBQFT). We find that an even or odd symmetric eigenvector of the 2-D DFT will also be an eigenvector of the DQFT and the DRBQFT. Moreover, both the DQFT and the DRBQFT have 8 eigenspaces, which correspond to the eigenvalues of 1, -1, i, -i, j, -j, k, and -k. We also use the derived eigenvectors to fractionalize the DQFT and the DRBQFT and define the discrete fractional quaternion transform and the discrete fractional reduced biquaternion Fourier transform.