论文标题
块循环四元素矩阵的对角线和四量张量的T产品的快速计算
Block diagonalization of block circulant quaternion matrices and the fast calculation for T-product of quaternion tensors
论文作者
论文摘要
在颜色图像和灰色视频处理中基于T产品的真实张量方法取得了巨大成功，在彩色视频处理中建立了基于T-产物的Quaternion张量方法，这遇到了挑战，这是块循环循环Quaternion矩阵的块对角线。在本文中，我们表明离散的傅立叶矩阵$ \ mathbf {f_p} $不能对角线$ p \ times p $ p $ cripculant quaternion矩阵，统一的quaternion矩阵$ \ mathbf {f_p} $ \ mathbf {f_p}（1+ \ Mathbf {j}）/\ sqrt {2} $带有$ \ mathbf {j} $是Quaternion elgebra的假想单位。此外，我们建立了足够和必要的条件，以使循环季节矩阵的对角线矩阵的统一季节矩阵，这表明在季节域中实现了循环四元素矩阵的对角线化太难了。我们转向八元结构域，以实现循环季节矩阵的对角线化。 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循环四元素的大小$ p \ times p $，以$ o（p \ log p）$的费用通过快速傅立叶变换（FFT）；和统一矩阵$ \ MATHBF {F_P} \ MATHBF {P} \ otimes \ MathBf {I_M} $和$ \ \ \ \ \ Mathbf {f_p} \ Mathbf {p} NP $，以FFT为$ O（MNP \ Log P）$的费用。结果，我们提出了一种快速算法，以通过FFTS计算$ m \ times n \ times n \ times p $和$ n \ times s \ times s \ times p $三阶四元素张量，费用为$ o（mnsp）$，该$（mnsp）$几乎是$ 1/p $ by $ o（mnsp）的计算量的$ 1/p $。数值计算验证了复杂性分析的正确性。
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.
