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

TL;DR

This paper introduces a new quaternion total variation regularization model and a flexible quaternion GMRES method to efficiently solve ill-posed quaternion inverse problems, improving convergence and restoration quality.

Contribution

It proposes a novel quaternion regularization model combined with a flexible GMRES algorithm and establishes improved convergence theory for faster solutions.

Findings

Outperforms state-of-the-art methods in iteration steps and CPU time.

Achieves higher quality in restored color images.

Provides a sharp residual norm upper bound for the proposed method.

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.