Block Diagonalization of Block Circulant Quaternion Matrices and the Fast Calculation for T-Product of Quaternion Tensors

With quaternion matrices and quaternion tensors being gradually used in the color image and color video processing, the block diagonalization of block circulant quaternion matrices has become a key issue in the establishment of T-product based methods for quaternion tensors. Out of this consideration, we aim at establishing a fast calculation approach for the block diagonalization of block circulant quaternion matrices with the help of the fast Fourier transform (FFT). We first show that the discrete Fourier matrix \(\mathbf {F_p}\) cannot diagonalize \(p\times p\) circulant quaternion matrices, nor can the unitary quaternion matrices \(\mathbf {F_p}\textbf{j}\) and \(\mathbf {F_p}(1+\textbf{j})/\sqrt{2}\) with \(\textbf{j}\) being an imaginary unit of quaternion algebra. Then we prove that the unitary octonion matrix \(\mathbf {F_p}\textbf{p}\) with \(\textbf{p}=\textbf{l},\textbf{il}\) or \((\textbf{l}+\textbf{il})/\sqrt{2}\) (\(\textbf{l}, \textbf{il}\) being imaginary units of octonion algebra) can diagonalize a circulant quaternion matrix of size \(p\times p\), which further means that a block circulant quaternion matrix of size \(mp\times np\) can be block diagonalized at the cost of \(O(mnp\log p)\) via the FFT. As one of applications, we give a fast algorithm to speed up the calculation of the T-product between \(m\times n\times p\) and \(n\times s\times p\) third-order quaternion tensors via FFTs, whose computational magnitude is almost 1/p of the original one. As another application, we propose an effective compression strategy for third-order quaternion tensors with a certain low-rankness. Simulations on the color image and color video compression demonstrate that our compression strategy with no QSVD involved, can achieve higher quality compression in terms of PSNR values at much less time costs, compared with the QSVD-based methods.