Block diagonalization of block circulant quaternion matrices and the fast calculation for T-product of quaternion tensors

With the great success of the T-product based real tensor methods in the color image and gray video processing, the establishment of T-product based quaternion tensor methods in the color video processing has encountered a challenge, which is the block diagonalization of block circulant quaternion matrices. In this paper, we show that the discrete Fourier matrix $\mathbf{F_p}$ cannot diagonalize $p\times p$ circulant quaternion matrices, nor can the unitary quaternion matrices $\mathbf{F_p}\mathbf{j}$ and $\mathbf{F_p}(1+\mathbf{j})/\sqrt{2}$ with $\mathbf{j}$ being an imaginary unit of quaternion algebra. Further, we establish sufficient and necessary conditions for a unitary quaternion matrix being a diagonalization matrix of circulant quaternion matrices, which shows that achieving the diagonalization of circulant quaternion matrices in the quaternion domain is too hard. We turn to the octonion domain for achieving the diagonalization of circulant quaternion matrices. We prove that the unitary octonion matrix $\mathbf{F_p}\mathbf{p}$ with $\mathbf{p}=\mathbf{l},\mathbf{il}$ or $(\mathbf{l}+\mathbf{il})/\sqrt{2}$ can diagonalize a circulant quaternion matrix of size $p\times p$, at the cost of $O(p\log p)$ via the fast Fourier transform (FFT); and unitary matrices $\mathbf{F_p}\mathbf{p}\otimes \mathbf{I_m}$ and $\mathbf{F_p}\mathbf{p}\otimes \mathbf{I_n}$ can block diagonalize a block circulant quaternion matrix of size $mp\times np$, at the cost of $O(mnp\log p)$ via the FFT. As a result, we propose a fast algorithm to calculate the T-product between $m\times n\times p$ and $n\times s\times p$ third-order quaternion tensors via FFTs, at the cost of $O(mnsp)$, which is almost $1/p$ of the computational magnitude of computing T-product by its definition. Numerical calculations verify the correctness of the complexity analysis.