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Low-Rank Reduced Biquaternion Tensor Ring Decomposition and Tensor Completion

Hui Luo, Xin Liu, Wei Liu, Yang Zhang

TL;DR

This work introduces the reduced biquaternion tensor ring (RBTR) framework and the RBTR-SVD algorithm to achieve low-storage, high-quality tensor representations. It then couples RBTR ranks with total variation regularization to form RBTR-TV, a novel tensor completion method designed for color images and videos from partial observations under $P_{\Omega}$. The authors demonstrate that RBTR-based representations reduce storage costs while maintaining reconstruction accuracy, and RBTR-TV delivers superior PSNR and RSE performance compared with several state-of-the-art baselines. The approach shows promise for efficient high-dimensional data completion, with potential impact in image/video processing and related applications where color-channel interactions and multi-way structure are critical.

Abstract

We define the reduced biquaternion tensor ring (RBTR) decomposition and provide a detailed exposition of the corresponding algorithm RBTR-SVD. Leveraging RBTR decomposition, we propose a novel low-rank tensor completion algorithm RBTR-TV integrating RBTR ranks with total variation (TV) regularization to optimize the process. Numerical experiments on color image and video completion tasks indicate the advantages of our method.

Low-Rank Reduced Biquaternion Tensor Ring Decomposition and Tensor Completion

TL;DR

This work introduces the reduced biquaternion tensor ring (RBTR) framework and the RBTR-SVD algorithm to achieve low-storage, high-quality tensor representations. It then couples RBTR ranks with total variation regularization to form RBTR-TV, a novel tensor completion method designed for color images and videos from partial observations under . The authors demonstrate that RBTR-based representations reduce storage costs while maintaining reconstruction accuracy, and RBTR-TV delivers superior PSNR and RSE performance compared with several state-of-the-art baselines. The approach shows promise for efficient high-dimensional data completion, with potential impact in image/video processing and related applications where color-channel interactions and multi-way structure are critical.

Abstract

We define the reduced biquaternion tensor ring (RBTR) decomposition and provide a detailed exposition of the corresponding algorithm RBTR-SVD. Leveraging RBTR decomposition, we propose a novel low-rank tensor completion algorithm RBTR-TV integrating RBTR ranks with total variation (TV) regularization to optimize the process. Numerical experiments on color image and video completion tasks indicate the advantages of our method.
Paper Structure (9 sections, 8 theorems, 85 equations, 4 figures, 3 tables, 2 algorithms)

This paper contains 9 sections, 8 theorems, 85 equations, 4 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduced biquaternion tensor decomposition
  4. Reduced biquaternion tensor completion
  5. Experiment
  6. Color image completion
  7. Color video completion
  8. Conclusion and future works
  9. Declarations

Key Result

Lemma 2.1

(Reduced biquaternion singular value decomposition RBSVD)pei2008eigenvalues With the notations in the formula RBMR, if the SVDs of $\mathbf{Q}_{c1}$ and $\mathbf{Q}_{c2}$ are in the following forms: then the SVD of ${\mathbf{Q}}$ is where ${\mathbf{U}} = \mathbf{U}_1 e_1 + \mathbf{U}_2 e_2$, $\boldsymbol{\Sigma} = \boldsymbol{\Sigma}_1 e_1 + \boldsymbol{\Sigma}_2 e_2$, ${\mathbf{V}} = \math

Figures (4)

  • Figure 1: Recovery performance of different methods at SR=20%
  • Figure 2: Recovery performance of different methods at SR=10%
  • Figure : RBTR-SVD
  • Figure : RBTR-TV

Theorems & Definitions (18)

  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • Definition 3.1
  • ...and 8 more