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Low-rank quaternion tensor completion for color video inpainting via a novel factorization strategy

Zhenzhi Qin, Zhenyu Ming, Defeng Sun, Liping Zhang

TL;DR

It is proved that the optimal low-rank approximation of a third-order quaternion tensor exists and some numerical experiments demonstrate the low-rankness of color videos.

Abstract

Recently, a quaternion tensor product named Qt-product was proposed, and then the singular value decomposition and the rank of a third-order quaternion tensor were given. From a more applicable perspective, we extend the Qt-product and propose a novel multiplication principle for third-order quaternion tensor named gQt-product. With the gQt-product, we introduce a brand-new singular value decomposition for third-order quaternion tensors named gQt-SVD and then define gQt-rank and multi-gQt-rank. We prove that the optimal low-rank approximation of a third-order quaternion tensor exists and some numerical experiments demonstrate the low-rankness of color videos. So, we apply the low-rank quaternion tensor completion to color video inpainting problems and present alternating least-square algorithms to solve the proposed low gQt-rank and multi-gQt-rank quaternion tensor completion models. The convergence analyses of the proposed algorithms are established and some numerical experiments on various color video datasets show the high recovery accuracy and computational efficiency of our methods.

Low-rank quaternion tensor completion for color video inpainting via a novel factorization strategy

TL;DR

It is proved that the optimal low-rank approximation of a third-order quaternion tensor exists and some numerical experiments demonstrate the low-rankness of color videos.

Abstract

Recently, a quaternion tensor product named Qt-product was proposed, and then the singular value decomposition and the rank of a third-order quaternion tensor were given. From a more applicable perspective, we extend the Qt-product and propose a novel multiplication principle for third-order quaternion tensor named gQt-product. With the gQt-product, we introduce a brand-new singular value decomposition for third-order quaternion tensors named gQt-SVD and then define gQt-rank and multi-gQt-rank. We prove that the optimal low-rank approximation of a third-order quaternion tensor exists and some numerical experiments demonstrate the low-rankness of color videos. So, we apply the low-rank quaternion tensor completion to color video inpainting problems and present alternating least-square algorithms to solve the proposed low gQt-rank and multi-gQt-rank quaternion tensor completion models. The convergence analyses of the proposed algorithms are established and some numerical experiments on various color video datasets show the high recovery accuracy and computational efficiency of our methods.
Paper Structure (19 sections, 19 theorems, 174 equations, 5 figures, 2 tables, 4 algorithms)

This paper contains 19 sections, 19 theorems, 174 equations, 5 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Quaternions
  4. Quaternion matrix and quaternion tensor
  5. QDFT, gQt-Product and gQt-SVD
  6. QDFT: a new DFT of third-order quaternion tensor
  7. gQt-product: a new product between third-order quaternion tensors
  8. gQt-SVD: gQt-product based SVD of third-order quaternion tensor
  9. gQt-rank, low-rank optimal approximation, and multi-gQt-rank
  10. Quaternion Tensor Completion for Color Video Inpainting
  11. Low gQt-rank quaternion tensor completion model
  12. Solution method
  13. Convergence analysis for QRTC
  14. Low multi-gQt-rank tensor completion model
  15. Numerical Experiments
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $A\in{\mathbb{H}}^{n_1\times n_2}$ and $B\in{\mathbb{H}}^{n_2\times n_3}$, then (i)$(AB)^*=B^*A^*$, (ii)$(AB)^{-1}=B^{-1}A^{-1}$ if $A$ and $B$ are invertible, (iii)$(A^*)^{-1}=(A^{-1})^*$ if $A$ is invertible.

Figures (5)

  • Figure 1: The sampled frames in video and singular values.
  • Figure 2: The sampled frames in video and singular values.
  • Figure 3: Comparison of RSE result of TCTF and QRTC-1 for color video recovery on 15 videos, sample ratio $\rho=0.1,\ 0.3,\ 0.5$ and $0.7$. From RSE, our Qt-SVD based QRTC outperforms over t-SVD based TCTF. Moreover, the smaller of sample size, the performance of QRTC-1 is better from (a).
  • Figure 4: First frame of color video recovery using different algorithms ($\rho=0.3$).
  • Figure 5: First frame of color video recovery while sample ratio $\rho$ from 0.1 to 0.5 using different methods.

Theorems & Definitions (49)

  • Lemma 2.1
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Definition 2.1: Qt-product Qin2022
  • Lemma 3.1
  • proof
  • Definition 3.1: gQt-product
  • Remark 3.1
  • Remark 3.2
  • ...and 39 more