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.

翻译：本文的主要目标是提出一种新的四元数全变分正则化模型，用于求解线性病态四元数逆问题，该问题源于三维信号滤波或彩色图像处理。模型中的四元数全变分项通过协同全变分正则化表示，并采用四元数迭代重加权范数进行逼近。本文提出了一种新颖的柔性四元数广义最小残差法来快速求解该模型。建立了一个改进的收敛性理论，以获得四元数最小残差法（QGMRES）残差范数的尖锐上界。该收敛性理论也适用于预条件处理的QGMRES。数值实验表明，所提出的模型和算法在迭代步数、CPU时间以及复原彩色图像的质量指标方面均优于现有最先进方法。