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

Daochang Zhang, Yue Zhao, Jingqian Li, Dijana Mosic

TL;DR

The paper addresses efficient quaternion tensor computation by exploiting a quaternion $Z$-block circulant structure and a corresponding block-diagonalization that yields a multiplicative equivalence between QT-products and matrix products. It introduces the $\mathtt{bcirc_z}$-inv algorithm and proves that $\mathtt{bcirc_z}(\mathcal A)$ can be diagonalized via the third-mode DFT, enabling scalable quaternion tensor decompositions including QT-Polar, QT-PLU, QT-LU, and QT-SVD without restrictive diagonal assumptions. The approach is validated through extensive large-scale experiments, demonstrating faster and more stable inverses, and is applied to Tikhonov-regularized models and video rotation to show improved coherence and color fidelity. Overall, the work provides a rigorous, scalable framework for quaternion tensor analysis with practical impact on color image and video processing.

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.

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

TL;DR

The paper addresses efficient quaternion tensor computation by exploiting a quaternion -block circulant structure and a corresponding block-diagonalization that yields a multiplicative equivalence between QT-products and matrix products. It introduces the -inv algorithm and proves that can be diagonalized via the third-mode DFT, enabling scalable quaternion tensor decompositions including QT-Polar, QT-PLU, QT-LU, and QT-SVD without restrictive diagonal assumptions. The approach is validated through extensive large-scale experiments, demonstrating faster and more stable inverses, and is applied to Tikhonov-regularized models and video rotation to show improved coherence and color fidelity. Overall, the work provides a rigorous, scalable framework for quaternion tensor analysis with practical impact on color image and video processing.

Abstract

The motivation of this paper is twofold. First, we investigate the block-diagonalization of the -block circulant matrix , based on this block-diagonal structure, and develop the algorithm -inv for computing the inverse of . Second, we establish the equivalence between the QT-product of tensors and the product of the corresponding -block circulant matrices. Based on this equivalence and in combination with the algorithm -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 -block circulant matrices are provided. As applications, a series of quaternion tensor decompositions under the QT-product and their corresponding -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 and . 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.
Paper Structure (6 sections, 9 theorems, 137 equations, 3 figures, 5 tables, 4 algorithms)

This paper contains 6 sections, 9 theorems, 137 equations, 3 figures, 5 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and key lemmas
  3. Block-Diagonalization and Multiplicative Equivalence of Quaternion $Z$-Block Circulant Matrices
  4. Quaternion tensor decompositions via QT-product
  5. Large-scale experiments and Scalability analysis
  6. Applications in Tikhonov-Regularized Model and Video Rotation

Key Result

Theorem 3.2

Let $\mathcal{A}_{\mathbf{d}}$ and $\mathcal{A}_{\mathbf{c}}\in \mathbb C^{n_1\times n_2\times n_3}$ satisfy that $\mathcal{A}=\mathcal{A}_{\mathbf{d}} + \mathbf{j} \mathcal{A}_{\mathbf{c}}\in\mathbb{Q}^{{n_1\times n_2\times n_3}}.$ Denote that $\hat{\mathcal{A}}$ is the DFT of $\mathcal{A}$. Then,

Figures (3)

  • Figure 1: Performance Analysis of Polar Decomposition Algorithm
  • Figure 2: Performance Analysis of PLU Decomposition Algorithm
  • Figure 3: Scalability analysis of computing $\mathbf A^{-1}$ with two methods

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 3.1: Z-Block Circulant Matrix
  • Theorem 3.2
  • ...and 36 more