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A New Cross-Space Total Variation Regularization Model for Color Image Restoration with Quaternion Blur Operator

Zhigang Jia, Yuelian Xiang, Meixiang Zhao, Tingting Wu, Michael K. Ng

TL;DR

The paper addresses cross-channel color image deblurring by introducing a cross-space total variation (CSTV) model that couples HSV and RGB information through SVTV and CTV, respectively. It employs a quaternion blur operator to faithfully represent cross-channel degradation and derives both Euler–Lagrange and dual formulations, proving existence and uniqueness under injectivity for the proposed model. A fast, structure-preserving quaternion operator splitting algorithm (CSTV$_{QGMRES}$) is developed, with an L-surface method to balance regularization parameters across color spaces. Numerical experiments on symmetric and asymmetric blur scenarios show CSTV consistently outperforming state-of-the-art methods in PSNR, SSIM, MSE, and CIEde2000, demonstrating improved color fidelity and texture preservation with robust cross-channel deblurring performance.

Abstract

The cross-channel deblurring problem in color image processing is difficult to solve due to the complex coupling and structural blurring of color pixels. Until now, there are few efficient algorithms that can reduce color artifacts in deblurring process. To solve this challenging problem, we present a novel cross-space total variation (CSTV) regularization model for color image deblurring by introducing a quaternion blur operator and a cross-color space regularization functional. The existence and uniqueness of the solution are proved and a new L-curve method is proposed to find a balance of regularization terms on different color spaces. The Euler-Lagrange equation is derived to show that CSTV has taken into account the coupling of all color channels and the local smoothing within each color channel. A quaternion operator splitting method is firstly proposed to enhance the ability of color artifacts reduction of the CSTV regularization model. This strategy also applies to the well-known color deblurring models. Numerical experiments on color image databases illustrate the efficiency and effectiveness of the new model and algorithms. The color images restored by them successfully maintain the color and spatial information and are of higher quality in terms of PSNR, SSIM, MSE and CIEde2000 than the restorations of the-state-of-the-art methods.

A New Cross-Space Total Variation Regularization Model for Color Image Restoration with Quaternion Blur Operator

TL;DR

The paper addresses cross-channel color image deblurring by introducing a cross-space total variation (CSTV) model that couples HSV and RGB information through SVTV and CTV, respectively. It employs a quaternion blur operator to faithfully represent cross-channel degradation and derives both Euler–Lagrange and dual formulations, proving existence and uniqueness under injectivity for the proposed model. A fast, structure-preserving quaternion operator splitting algorithm (CSTV) is developed, with an L-surface method to balance regularization parameters across color spaces. Numerical experiments on symmetric and asymmetric blur scenarios show CSTV consistently outperforming state-of-the-art methods in PSNR, SSIM, MSE, and CIEde2000, demonstrating improved color fidelity and texture preservation with robust cross-channel deblurring performance.

Abstract

The cross-channel deblurring problem in color image processing is difficult to solve due to the complex coupling and structural blurring of color pixels. Until now, there are few efficient algorithms that can reduce color artifacts in deblurring process. To solve this challenging problem, we present a novel cross-space total variation (CSTV) regularization model for color image deblurring by introducing a quaternion blur operator and a cross-color space regularization functional. The existence and uniqueness of the solution are proved and a new L-curve method is proposed to find a balance of regularization terms on different color spaces. The Euler-Lagrange equation is derived to show that CSTV has taken into account the coupling of all color channels and the local smoothing within each color channel. A quaternion operator splitting method is firstly proposed to enhance the ability of color artifacts reduction of the CSTV regularization model. This strategy also applies to the well-known color deblurring models. Numerical experiments on color image databases illustrate the efficiency and effectiveness of the new model and algorithms. The color images restored by them successfully maintain the color and spatial information and are of higher quality in terms of PSNR, SSIM, MSE and CIEde2000 than the restorations of the-state-of-the-art methods.
Paper Structure (25 sections, 2 theorems, 90 equations, 17 figures, 6 tables, 2 algorithms)

This paper contains 25 sections, 2 theorems, 90 equations, 17 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cross-space total variation regularization model for color image restoration
  4. Fidelity functional with quaternion blur operator
  5. Cross-space total variation regularization functional
  6. A new color image restoration model
  7. Fast and stable algorithm
  8. Quaternion operator splitting method for $\mathbf{u}$-subproblem
  9. Augmented Lagrangian method for $(\mathbf{w},{\bf v})$-subproblem
  10. A new fast and stable algorithm based on quaternion operator splitting
  11. Numerical Experiments
  12. Symmetric cross-channel deblurring problem
  13. Asymmetric cross-channel deblurring problem
  14. Analysis of cross-channel deblurring ability
  15. Conclusion
  16. ...and 10 more sections

Key Result

Theorem 1

With the above notation, the cross-channel blurring process e:blurmodel0 is equivalent to where quaternion operator ${\bf Q}=\Re^{-1}(Q)$ and quaternion function ${{\bf r}}(x,y)=\Re_c^{-1}(R\star \Re_c(\mathbf{u}(x,y))$. Especially, one quaternion operator satisfying e:blurmodel0q is ${\bf Q}=Q_0+Q_1\mathbf{i}+Q_2\mathbf{j}+Q_3\mathbf{k}$ with where $w_{ij}$ and $K_{ij}$ are given in (e:KW).

Figures (17)

  • Figure 1: A clear picture of the real-world object (Cattleya$\times$hybrida H.J.Veitch) and the simulations of motional blur (length=$15$, angle=$45^\circ$, symmetric) by using $\hat{K}$ and ${\bf Q}$.
  • Figure 2: A clear picture of the real-world object (Cattleya$\times$hybrida H.J.Veitch) and the simulations of motional blur (length=$15$, angle=$45^\circ$, asymmetric) by using $\hat{K}$ and ${\bf Q}$.
  • Figure 3: The values of PSNR and SSIM for the restored image ‘statues’ in different $\lambda_1, \lambda_2$ conditions.
  • Figure 4: The values of CIEde2000 for the restored image ‘statues’ in different $\alpha_1$ and $\alpha_2$ conditions.
  • Figure 5: Ground truth images.
  • ...and 12 more figures

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • Remark 2
  • proof