As a new color image representation tool, quaternion has achieved excellent results in the color image processing, because it treats the color image as a whole rather than as a separate color space component, thus it can make full use of the high correlation among RGB channels. Recently, low-rank quaternion matrix completion (LRQMC) methods have proven very useful for color image inpainting. In this article, we propose three novel LRQMC methods based on three quaternion-based bilinear factor (QBF) matrix norm minimization models. Specifically, we define quaternion double Frobenius norm (Q-DFN), quaternion double nuclear norm (Q-DNN), and quaternion Frobenius/nuclear norm (Q-FNN), and then show their relationship with quaternion-based matrix Schatten-$p$ (Q-Schatten-$p$) norm for certain $p$ values. The proposed methods can avoid computing quaternion singular value decompositions (QSVD) for large quaternion matrices, and thus can effectively reduce the calculation time compared with existing (LRQMC) methods. Furthermore, we also analyze the convergence of the algorithms, and introduce a rank estimation method for the quaternion matrix. The experimental results demonstrate the superior performance of the proposed methods over some state-of-the-art low-rank (quaternion) matrix completion methods.

中文翻译：

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基于四元数的双线性因子矩阵范数最小化彩色图像修复

四元数作为一种新的彩色图像表示工具，在彩色图像处理中取得了很好的效果，因为它把彩色图像作为一个整体而不是一个单独的颜色空间分量来处理，因此可以充分利用RGB通道之间的高相关性. 最近，低秩四元数矩阵补全（LRQMC）方法已被证明对彩色图像修复非常有用。在本文中，我们基于三个基于四元数的双线性因子 (QBF) 矩阵范数最小化模型提出了三种新颖的 LRQMC 方法。具体来说，我们定义四元数双 Frobenius 范数（Q-DFN）、四元数双核范数（Q-DNN）和四元数 Frobenius/核范数（Q-FNN），然后展示它们与基于四元数的矩阵 Schatten-$p$ (Q-Schatten-$p$) 特定标准 $p$值。所提出的方法可以避免计算大型四元数矩阵的四元数奇异值分解（QSVD），因此与现有（LRQMC）方法相比，可以有效地减少计算时间。此外，我们还分析了算法的收敛性，并介绍了四元数矩阵的秩估计方法。实验结果证明了所提出的方法优于一些最先进的低秩（四元数）矩阵完成方法的性能。