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Nonconvex Robust Quaternion Matrix Completion for Imaging Processing

Baohua Huang, Jiakai Chen, Wen Li

TL;DR

A new nonconvex robust QMC model is proposed, in which the nonconvex MCP function and the quaternion $L_p-norm are used to enhance the low-rankness and sparseness of the low-rank term and sparse term, respectively and an alternating direction method of multipliers (ADMM) algorithm is developed to solve the model.

Abstract

One of the tasks in color image processing and computer vision is to recover clean data from partial observations corrupted by noise. To this end, robust quaternion matrix completion (QMC) has recently attracted more attention and shown its effectiveness, whose convex relaxation is to minimize the quaternion nuclear norm plus the quaternion $L_1$-norm. However, there is still room to improve due to the convexity of the convex surrogates. This paper proposes a new nonconvex robust QMC model, in which the nonconvex MCP function and the quaternion $L_p$-norm are used to enhance the low-rankness and sparseness of the low-rank term and sparse term, respectively. An alternating direction method of multipliers (ADMM) algorithm is developed to solve the proposed model and its convergence is given. Moreover, a novel nonlocal-self-similarity-based nonconvex robust quaternion completion method is proposed to handle large-scale data. Numerical results on color images and videos indicate the advantages of the proposed method over some existing ones.

Nonconvex Robust Quaternion Matrix Completion for Imaging Processing

TL;DR

A new nonconvex robust QMC model is proposed, in which the nonconvex MCP function and the quaternion $L_p-norm are used to enhance the low-rankness and sparseness of the low-rank term and sparse term, respectively and an alternating direction method of multipliers (ADMM) algorithm is developed to solve the model.

Abstract

One of the tasks in color image processing and computer vision is to recover clean data from partial observations corrupted by noise. To this end, robust quaternion matrix completion (QMC) has recently attracted more attention and shown its effectiveness, whose convex relaxation is to minimize the quaternion nuclear norm plus the quaternion -norm. However, there is still room to improve due to the convexity of the convex surrogates. This paper proposes a new nonconvex robust QMC model, in which the nonconvex MCP function and the quaternion -norm are used to enhance the low-rankness and sparseness of the low-rank term and sparse term, respectively. An alternating direction method of multipliers (ADMM) algorithm is developed to solve the proposed model and its convergence is given. Moreover, a novel nonlocal-self-similarity-based nonconvex robust quaternion completion method is proposed to handle large-scale data. Numerical results on color images and videos indicate the advantages of the proposed method over some existing ones.

Paper Structure

This paper contains 12 sections, 8 theorems, 69 equations, 6 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The NRQMC model and algorithm
  4. The motivation
  5. The proposed model and algorithm
  6. Convergence analysis
  7. NSS-based method for large-scale color image and video inpainting
  8. Numerical experiments
  9. Parameters Setting
  10. Color Images
  11. Color Videos
  12. Concluding remarks

Key Result

Proposition 1

For $\mathbf{X},\bm{\Delta}\mathbf{X}\in\mathbb{Q}^{n_1\times n_2}$, it holds $h'(\mathbf{X};\bm{\Delta}\mathbf{X})= \langle\nabla h(\mathbf{X})$, $\bm{\Delta}\mathbf{X}\rangle$. If $h$ is continuous differentiable, then $h(\mathbf{X}+\bm{\Delta}\mathbf{X})=h(\mathbf{X})+\langle\nabla h(\mathbf{X})$

Figures (6)

  • Figure 1: Comparison of the quaternion matrix rank, the QNN and the MCP approximation rank function given by (\ref{['3.2']}).
  • Figure 2: Performance of the proposed methods with different parameters on Lena under $SR=0.2$.
  • Figure 3: Recovery results by different methods on House, Peppers, and Lena (from top to bottle) under $SR=0.8$ and $\gamma = 0.2$. (a) Observed. (b) TNN. (c) LRQA-G. (d) QNN. (e) PQMC. (f) NRQMC. (g) PQMC-NSS2D. (h) NRQMC-NSS2D. (i) Original.
  • Figure 4: Recovery results by different methods on Kodak image12 under $SR=0.5$ and $\gamma = 0.1$. (a) Observed. (b) TNN. (c) LRQA-G. (d) QNN. (e) PQMC. (f) NRQMC. (g) PQMC-NSS2D. (h) NRQMC-NSS2D. (i) Original.
  • Figure 5: Recovery results by different methods on Akiyo, Foreman, Salesman, and Hall (from top to bottle) under $SR=0.3$. (a) Observed. (b) HaLRTC. (c) SiLRTC. (d) TMac-inc. (e) TMac-dec. (f) LRQTC. (g) PQMC-NSS3D. (h) NRQMC-NSS3D. (i) Original.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Definition 1
  • Definition 2: qi2020quaternionchen2022color
  • Proposition 1: qi2020quaternion
  • Proposition 2: qi2020quaternion
  • Theorem 1
  • Remark 1
  • Theorem 2: Quaternion thresholding operator for MCP penalty
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • ...and 1 more