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

TL;DR

This paper explores the block-diagonalization of quaternion $z$-block circulant matrices, establishes their equivalence with tensor products, and applies these findings to develop decompositions and algorithms for large-scale quaternion tensor computations, including applications in video rotation.

Contribution

It introduces new algorithms for block-diagonalization and inversion of quaternion $z$-block circulant matrices, and establishes their equivalence with tensor products, enabling advanced tensor decompositions and applications.

Findings

Developed the $ exttt{bcirc_z}$-inv algorithm for matrix inversion.

Established the equivalence between QT-product of tensors and matrix product.

Demonstrated stability and accuracy of quaternion tensor decompositions in video rotation 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.