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Quaternion Nuclear Norm minus Frobenius Norm Minimization for color image reconstruction

Yu Guo, Guoqing Chen, Tieyong Zeng, Qiyu Jin, Michael Kwok-Po Ng

TL;DR

Demonstrating versatility and efficacy, the QNMF regularizer excels in various color low-level vision tasks, including denoising, deblurring, inpainting, and random impulse noise removal, achieving state-of-the-art results.

Abstract

Color image restoration methods typically represent images as vectors in Euclidean space or combinations of three monochrome channels. However, they often overlook the correlation between these channels, leading to color distortion and artifacts in the reconstructed image. To address this, we present Quaternion Nuclear Norm Minus Frobenius Norm Minimization (QNMF), a novel approach for color image reconstruction. QNMF utilizes quaternion algebra to capture the relationships among RGB channels comprehensively. By employing a regularization technique that involves nuclear norm minus Frobenius norm, QNMF approximates the underlying low-rank structure of quaternion-encoded color images. Theoretical proofs are provided to ensure the method's mathematical integrity. Demonstrating versatility and efficacy, the QNMF regularizer excels in various color low-level vision tasks, including denoising, deblurring, inpainting, and random impulse noise removal, achieving state-of-the-art results.

Quaternion Nuclear Norm minus Frobenius Norm Minimization for color image reconstruction

TL;DR

Demonstrating versatility and efficacy, the QNMF regularizer excels in various color low-level vision tasks, including denoising, deblurring, inpainting, and random impulse noise removal, achieving state-of-the-art results.

Abstract

Color image restoration methods typically represent images as vectors in Euclidean space or combinations of three monochrome channels. However, they often overlook the correlation between these channels, leading to color distortion and artifacts in the reconstructed image. To address this, we present Quaternion Nuclear Norm Minus Frobenius Norm Minimization (QNMF), a novel approach for color image reconstruction. QNMF utilizes quaternion algebra to capture the relationships among RGB channels comprehensively. By employing a regularization technique that involves nuclear norm minus Frobenius norm, QNMF approximates the underlying low-rank structure of quaternion-encoded color images. Theoretical proofs are provided to ensure the method's mathematical integrity. Demonstrating versatility and efficacy, the QNMF regularizer excels in various color low-level vision tasks, including denoising, deblurring, inpainting, and random impulse noise removal, achieving state-of-the-art results.
Paper Structure (17 sections, 5 theorems, 51 equations, 10 figures, 6 tables, 3 algorithms)

This paper contains 17 sections, 5 theorems, 51 equations, 10 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Quaternion Basics
  4. Quaternion Low Rank Methods
  5. Quaternion Nuclear Norm minus Frobenius Norm Minimization
  6. QNMF model
  7. QNMF for image linear inverse problem
  8. QNMF for matrix completion
  9. QNMF for robust PCA
  10. Experiments
  11. Experiment Setting
  12. Color image denoising
  13. Real image denoising
  14. Color image debluring
  15. Matrix completion and Robust PCA
  16. ...and 2 more sections

Key Result

Lemma 2.1

(QSVD zhang1997quaternions) Given a quaternion matrix $\dot{\mathbf{X}} \in \mathbb{ Q }^{m\times n}$ with rank $r$. There are two unitary quaternion matrices $\dot{\mathbf{U}} \in \mathbb{ Q }^{m\times m}$ and $\dot{\mathbf{V}} \in \mathbb{ Q }^{n\times n}$ satisfying $\dot{\mathbf{X}} = \dot{\mat

Figures (10)

  • Figure 1: Comparison of color image denoising algorithms with different strategies in terms of color distortion and artifacts. The images are $45\times45$ patches, sourced from the Kodak dataset, and are corrupted by noise with $\sigma=60$.
  • Figure 2: Singular value shrinkage diagrammed.
  • Figure 3: CSet12.
  • Figure 4: Comparison of visual results for color image denoising. The first two rows of images are from CSet12. The last two rows of images are from the McMaster dataset.
  • Figure 5: Comparison of visual results for color image denoising. All images are from the Kodak dataset.
  • ...and 5 more figures

Theorems & Definitions (8)

  • Lemma 2.1
  • Theorem 3.1
  • proof : Proof
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • proof : Proof
  • proof