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Singular value decompositions of third-order reduced biquaternion tensors

Cui-E Yu, Xin Liu, Yang Zhang

TL;DR

This paper develops algorithms for computing the singular value decomposition (SVD) of a third-order reduced biquaternion tensor via a new Ht-product and defines the Moore-Penrose inverse of a third-order reduced biquaternion tensor and develops its characterizations.

Abstract

In this paper, we introduce the applications of third-order reduced biquaternion tensors in color video processing. We first develop algorithms for computing the singular value decomposition (SVD) of a third-order reduced biquaternion tensor via a new Ht-product. As theoretical applications, we define the Moore-Penrose inverse of a third-order reduced biquaternion tensor and develop its characterizations. In addition, we discuss the general (or Hermitian) solutions to reduced biquaternion tensor equation $\mathcal{A}\ast_{Ht} \mathcal{X}=\mathcal{B}$ as well as its least-square solution. Finally, we compress the color video by this SVD, and the experimental data shows that our method is faster than the compared scheme.

Singular value decompositions of third-order reduced biquaternion tensors

TL;DR

This paper develops algorithms for computing the singular value decomposition (SVD) of a third-order reduced biquaternion tensor via a new Ht-product and defines the Moore-Penrose inverse of a third-order reduced biquaternion tensor and develops its characterizations.

Abstract

In this paper, we introduce the applications of third-order reduced biquaternion tensors in color video processing. We first develop algorithms for computing the singular value decomposition (SVD) of a third-order reduced biquaternion tensor via a new Ht-product. As theoretical applications, we define the Moore-Penrose inverse of a third-order reduced biquaternion tensor and develop its characterizations. In addition, we discuss the general (or Hermitian) solutions to reduced biquaternion tensor equation as well as its least-square solution. Finally, we compress the color video by this SVD, and the experimental data shows that our method is faster than the compared scheme.
Paper Structure (7 sections, 19 theorems, 63 equations, 3 figures, 2 tables, 3 algorithms)

This paper contains 7 sections, 19 theorems, 63 equations, 3 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. The Ht-product of third-order tensors over $\mathbb{H}_c$
  3. The Ht-SVD of third-order tensors over $\mathbb{H}_c$
  4. The MP inverses of reduced biquaternion tensors
  5. Reduced biquaternion tensor equation $\mathcal{A} \ast_{Ht} \mathcal{X}=\mathcal{B}$
  6. Applications in color video processing
  7. Declarations

Key Result

Theorem 2.3

Let $\mathcal{A} \in \mathbb{H}_c^{n_1\times n_2 \times n_3}, \ \mathcal{B} \in \mathbb{H}_c^{n_2\times n_4 \times n_3}$ and $\mathcal{C} \in \mathbb{H}_c^{n_1\times n_4 \times n_3}$ with the corresponding DFTs $\widehat{\mathcal{A}}, \ \widehat{\mathcal{B}}$ and $\widehat{\mathcal{C}}$. Then $\mat

Figures (3)

  • Figure 1: PSNRs of the rank-k (k=10, 20, 50) approximations to DO01_013 by Ht-SVD.
  • Figure 2: Example of Deblurring Estimation for Three Blurred Frames in Video 'Tram'.
  • Figure 3: Example of Deblurring Estimation for Three Blurred Frames in Video 'Butterfly'.

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 18 more