On the Block-Diagonalization and Multiplicative Equivalence of Quaternion $Z$-Block Circulant Matrices with their Applications

Abstract: The motivation of this paper is twofold. First, we investigate the block-diagonalization of the $z$-block circulant matrix $\mathtt{bcirc_z}(\mathcal A)$, based on this block-diagonal structure, and develop the algorithm $\mathtt{bcirc_z}$-inv for computing the inverse of $\mathtt{bcirc_z}(\mathcal A)$. Second, we establish the equivalence between the QT-product of tensors and the product of the corresponding $z$-block circulant matrices. Based on this equivalence and in combination with the algorithm $\mathtt{bcirc_z}$-inv, large-scale tests and scalability analysis of the Tikhonov-regularized model are conducted. As a by-product of the analysis, some relevant and straightforward properties of the quaternion $z$-block circulant matrices are provided. As applications, a series of quaternion tensor decompositions under the QT-product and their corresponding $z$-block circulant matrices decompositions are obtained, including the QT-Polar decomposition, the QT-PLU decomposition, and the QT-LU decomposition. Meanwhile, the QT-SVD is rederived based on the relation between $\mathcal A$ and $\mathtt{bcirc_z}(\mathcal A)$. Furthermore, we develop corresponding algorithms and present several large-scale tests and scalability analysis. In addition, applications in video rotation are presented to evaluate several rotation strategies based on the QT-Polar decomposition, which shows the decomposition remains stable and inter-frame consistent while accurately maintaining color reproduction.