Singular value decomposition of third order quaternion tensors

doi:10.1016/j.aml.2021.107597

Applied Mathematics Letters

Volume 123, January 2022, 107597

https://doi.org/10.1016/j.aml.2021.107597 Get rights and content

Abstract

A novel tensor product for third order quaternion tensors is proposed. With this tensor product, we further define the singular value decomposition and the rank of a third order quaternion tensor. It is also proved that the best rank- $k$ approximation of a third order quaternion tensor exists. On the other hand, note that a color video can be expressed as a third order quaternion tensor, the proposed singular value decomposition thus opens up a new way to analyze the structure of color video. Finally, some experiments are performed to demonstrate the low-rankness of color videos.

Introduction

Tensors, as higher order generalizations of vectors and matrices, have wide applications in signal processing, computer vision, graph analysis, to name a few. There are three popular tensor factorization methods, namely, the CANDECOMP/PARAFAC (CP) decomposition [1], [2], the Tucker decomposition [3] and the tensor singular value decomposition (t-SVD) [4]. However, the three mentioned techniques are not useful for the classic color video inpainting problems. In specific, the traditional t-SVD is only applicable to third order tensors, while a color video is described as a fourth order tensor. Moreover, the computational complexity of CP decomposition is NP-hard, and the unfolding operation in Tucker decomposition will destroy the original multi-way structure of the data. Also note that although there are a number of studies applying quaternion technique to color image representation [5], [6], [7], [8], the work focusing on color video inpainting problem is still rather rare. A recent work on color video inpainting contributed by Jia, Jin, Ng and Zhao can be referred to on the arXiv website.¹ But their method requires that all of frontal slices should miss pixels at the same positions. All these dilemmas motivate us to design a new type of tensor factorization strategy to better exploit the structure of color videos, and hopefully, the accompanied technique can be applied to color video inpainting problems with no extra restriction.

The set $Q$ of quaternions is a four-dimensional linear space over real number field $R$ with an ordered basis, denoted by $1$ , $i$ , $j$ and $k$ . Here $i$ , $j$ and $k$ are three imaginary units with the following multiplication laws: $i^{2} = j^{2} = k^{2} = - 1, i j = - j i = k, j k = - k j = i, k i = - i k = j .$ In this paper, we first introduce a product for third order quaternion tensor (Qt-product), which is the generalization of t-product in [4]. Based on Qt-product, we propose a new SVD of third order quaternion tensor (Qt-SVD). This makes it possible to analyze the structure of color video without missing the original multi-way structure of the data. In addition, the rank of a third order quaternion tensor (denoted as Qt-rank), the identity tensor and the unitary tensor are also defined. It is notable that the best rank- $k$ approximation in this sense is attainable, which implies the coincide between the Qt-rank and its approximation. Experiments on several color videos are presented in Section 3, and they verify that the proposed Qt-SVD can embody the low-rankness of natural color videos.

Section snippets

Qt-product and Qt-SVD for third order quaternion tensor

One major contribution of this paper is that we propose a novel SVD of third order quaternion tensor, called the Qt-SVD. At beginning, we define a new type of third order quaternion tensor product (Qt-product), which is an extension of t-product mentioned in [4].

Before proceeding, we present some notations here. Scalars, vectors, matrices and third order tensors are denoted as lowercase letters ( $a, b, \dots$ ), bold-case lowercase letters ( $a, b, \dots$ ), capital letters ( $A, B, \dots$ ) and Euler script letters ( $A, B, \dots$

Applications

To embody the numerical applications of the proposed Qt-SVD theory, we first investigate the Qt-rank of third order quaternion tensors generated by color videos, and show that the (approximated) low-Qt-rankness is actually an inherent property of many color videos. In specific, the red, green and blue pixel values are set as the $i, j$ and $k$ parts of the quaternion tensor, respectively. Thus, the real part of such tensor vanishes. The tested videos are “DO01_013”, “DO01_055”, “M07_058” and

Acknowledgments

The authors would like to thank the anonymous referees for their valuable suggestions which helped to significantly improve the content of this paper. This work is supported by the National Natural Science Foundation of China (Grant No. 12171271,11771244).

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Citation Excerpt :
Thus, to handle color videos more naturally with quaternions, we defined quaternion tensors in our previous work [11]. In addition, to better make low-rank approximations for color videos, the authors in [12] generalized the tensor singular value decomposition (t-SVD) [13] to the quaternion domain and proposed a singular value decomposition approach for third-order quaternion tensors (Qt-SVD). Following the important property, the same as t-product [13] based on, that any complex circulant matrix can be diagonalized by the normalized Discrete Fourier Transformation (DFT) matrix, the authors in [12] designed a Qt-product by rewriting the quaternion tensor as two complex tensors.
A flexible transform-based tensor product named $★_{QT}$ -product for $L$ th-order ( $L \geq 3$ ) quaternion tensors is proposed. Based on the $★_{QT}$ -product, we define the corresponding singular value decomposition named TQt-SVD and the rank named TQt-rank of the $L$ th-order ( $L \geq 3$ ) quaternion tensor. Furthermore, with orthogonal quaternion transformations, the TQt-SVD can provide the best TQt-rank- $s$ approximation of any $L$ th-order ( $L \geq 3$ ) quaternion tensor. On the other side, since a color pixel with RGB channels can be well encoded as a pure quaternion, the proposed TQt-SVD, therefore, becomes a new mathematical tool for many color image processing tasks. In the experiments, we have verified the effectiveness of the proposed TQt-SVD in the application of the best TQt-rank- $s$ approximation for color videos represented by third-order quaternion tensors.
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Higher-order singular value decomposition (HOSVD) is one of the most efficient tensor decomposition techniques. It has the salient ability to represent high-dimensional data and extract features. On the other hand, in more recent years, the quaternion has proven to be a very suitable tool for color pixel representation as it can well preserve cross-channel correlation of color channels. Motivated by the advantages of the HOSVD and the quaternion tool, in this paper, we generalize the HOSVD to the quaternion domain and define quaternion-based HOSVD (QHOSVD). Due to the non-commutability of quaternion multiplication, QHOSVD is not a trivial extension of the HOSVD. They have similar but different calculation procedures. Theoretically, QHOSVD is a proper tensor generalization of the quaternion singular value decomposition (QSVD), and a proper quaternion generalization of the standard HOSVD. From the application point of view, the defined QHOSVD can be widely used in various visual data processing with color pixels. As examples, in this paper, we present two applications of the defined QHOSVD in color image processing—multi-focus color image fusion and color image denoising. The experimental results on the two applications respectively demonstrate the competitive performance of the proposed methods over some existing ones.

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Singular value decomposition of third order quaternion tensors

Abstract

Introduction

Section snippets

Qt-product and Qt-SVD for third order quaternion tensor

Applications

Acknowledgments

Linear Algebra Appl.

Linear Algebra Appl.

Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart–Young” decomposition

Psychometrika

Some mathematical notes on three-mode factor analysis

Psychometrika

Hypercomplex fourier transforms of color images

IEEE Trans. Image Process.