Block Diagonalization of Quaternion Circulant Matrices with Applications

TL;DR

This paper introduces a novel block-diagonalization method for quaternion circulant matrices using a permuted discrete quaternion Fourier transform, enabling efficient inversion and tensor decomposition with applications in quaternion signal processing.

Contribution

It demonstrates that quaternion circulant matrices can be block-diagonalized into 1x1 and 2x2 blocks, unlike complex circulant matrices, and applies this to quaternion tensor SVD and signal processing.

Findings

Efficient inversion of quaternion circulant matrices achieved.

Block-diagonalization enables quaternion tensor SVD.

Application to color video tensor processing validated.

Abstract

It is well-known that a complex circulant matrix can be diagonalized by a discrete Fourier matrix with imaginary unit $\mathtt{i}$. The main aim of this paper is to demonstrate that a quaternion circulant matrix cannot be diagonalized by a discrete quaternion Fourier matrix with three imaginary units $\mathtt{i}$, $\mathtt{j}$ and $\mathtt{k}$. Instead, a quaternion circulant matrix can be block-diagonalized into 1-by-1 block and 2-by-2 block matrices by permuted discrete quaternion Fourier transform matrix. With such a block-diagonalized form, the inverse of a quaternion circulant matrix can be determined efficiently similar to the inverse of a complex circulant matrix. We make use of this block-diagonalized form to study quaternion tensor singular value decomposition of quaternion tensors where the entries are quaternion numbers. The applications including computing the inverse of a quaternion circulant matrix, and solving quaternion Toeplitz system arising from linear prediction of quaternion signals are employed to validate the efficiency of our proposed block diagonalized results. A numerical example of color video as third-order quaternion tensor is employed to validate the effectiveness of quaternion tensor singular value decomposition.