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Flexible Quaternion Generalized Minimal Residual Method for Ill-Posed Quaternion Inverse Problems

Xuan Liu, Zhigang Jia, Xiaoqing Jin

TL;DR

This work addresses ill-posed quaternion inverse problems that arise in 3D signal processing and color image restoration by introducing a quaternion total variation (QTV) regularization framework. The authors develop a quaternion collaboratively weighted approach (QIRN) to reformulate QTV into a quaternion Tikhonov problem and solve it efficiently with a flexible quaternion generalized minimal residual method (FQGMRES), accompanied by an improved convergence theory that provides sharp residual bounds. They establish left, right, and flexible preconditioning strategies, along with a robust theoretical basis using quaternion polynomials and minimal polynomials to guarantee convergence. Numerical experiments demonstrate that the proposed QTV-FQGMRES framework outperforms state-of-the-art methods in iteration count, computational time, and color image restoration quality, highlighting its practical impact for fast and accurate quaternion-based inverse problems.

Abstract

The main goal of this paper is to propose a new quaternion total variation regularization model for solving linear ill-posed quaternion inverse problems, which arise from three-dimensional signal filtering or color image processing. The quaternion total variation term in the model is represented by collaborative total variation regularization and approximated by a quaternion iteratively reweighted norm. A novel flexible quaternion generalized minimal residual method is presented to quickly solve this model. An improved convergence theory is established to obtain a sharp upper bound of the residual norm of quaternion minimal residual method (QGMRES). The convergence theory is also presented for preconditioned QGMRES. Numerical experiments indicate the superiority of the proposed model and algorithms over the state-of-the-art methods in terms of iteration steps, CPU time, and the quality criteria of restored color images.

Flexible Quaternion Generalized Minimal Residual Method for Ill-Posed Quaternion Inverse Problems

TL;DR

This work addresses ill-posed quaternion inverse problems that arise in 3D signal processing and color image restoration by introducing a quaternion total variation (QTV) regularization framework. The authors develop a quaternion collaboratively weighted approach (QIRN) to reformulate QTV into a quaternion Tikhonov problem and solve it efficiently with a flexible quaternion generalized minimal residual method (FQGMRES), accompanied by an improved convergence theory that provides sharp residual bounds. They establish left, right, and flexible preconditioning strategies, along with a robust theoretical basis using quaternion polynomials and minimal polynomials to guarantee convergence. Numerical experiments demonstrate that the proposed QTV-FQGMRES framework outperforms state-of-the-art methods in iteration count, computational time, and color image restoration quality, highlighting its practical impact for fast and accurate quaternion-based inverse problems.

Abstract

The main goal of this paper is to propose a new quaternion total variation regularization model for solving linear ill-posed quaternion inverse problems, which arise from three-dimensional signal filtering or color image processing. The quaternion total variation term in the model is represented by collaborative total variation regularization and approximated by a quaternion iteratively reweighted norm. A novel flexible quaternion generalized minimal residual method is presented to quickly solve this model. An improved convergence theory is established to obtain a sharp upper bound of the residual norm of quaternion minimal residual method (QGMRES). The convergence theory is also presented for preconditioned QGMRES. Numerical experiments indicate the superiority of the proposed model and algorithms over the state-of-the-art methods in terms of iteration steps, CPU time, and the quality criteria of restored color images.
Paper Structure (15 sections, 12 theorems, 76 equations, 2 figures, 3 tables, 6 algorithms)

This paper contains 15 sections, 12 theorems, 76 equations, 2 figures, 3 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quaternion matrices
  4. Quaternion systems and real counterpart of quaternion matrices
  5. Collaborative total variation
  6. Preconditioned Quaternion Generalized Minimal Residual Method
  7. Improved convergence theory of QGMRES
  8. Left-Preconditioned QGMRES
  9. Right-Preconditioned QGMRES
  10. Flexible QGMRES
  11. Quaternion Total Variation Regularization Model
  12. Quaternion collaborative total variation regularization
  13. Quaternion iteratively reweighted norm method
  14. Numerical Experiments
  15. Conclusion

Key Result

Theorem 3.5

Assume that $\boldsymbol{\mathcal{A}}\in \mathbb{Q}^{n \times n}$ is a diagonalizable quaternion matrix. Let $\boldsymbol{\mathcal{A}}=\boldsymbol{\mathcal{X}}\boldsymbol{\Lambda} \boldsymbol{\mathcal{X}}^{-1}$and $\boldsymbol{\mathcal{X}}^{- 1}{\bf r}_0:=[\mathbf{x}_1,\mathbf{x}_2,\ldots,\mathbf{x} where Then, the residual norm is achieved by the $m$th step of QGMRES satisfies the inequality wh

Figures (2)

  • Figure 1: The relative residual errors of IRFGMRES and QTV-FQGMRES.
  • Figure 2: From left to right: orginal images; observed images; restored results by using IRFGMRES, SV-TV, QTV-FQGMRES.

Theorems & Definitions (29)

  • Definition 2.1: Inner product
  • Definition 2.2: Norm
  • Definition 3.1: Grade
  • Definition 3.2: Quaternion polynomial
  • Definition 3.3
  • Definition 3.4: Quaternion minimal polynomial
  • Theorem 3.5
  • Proof 1
  • Lemma 3.6
  • Corollary 1
  • ...and 19 more