The low-rank quaternion matrix approximation has been successfully applied in many applications involving signal processing and color image processing. However, the cost of quaternion models for generating low-rank quaternion matrix approximation is sometimes considerable due to the computation of the quaternion singular value decomposition (QSVD), which limits their application to real large-scale data. To address this deficiency, an efficient quaternion matrix CUR (QMCUR) method for low-rank approximation is suggested, which provides significant acceleration in color image processing. We first explore the QMCUR approximation method, which uses actual columns and rows of the given quaternion matrix, instead of the costly QSVD. Additionally, two different sampling strategies are used to sample the above-selected columns and rows. Then, the perturbation analysis is performed on the QMCUR approximation of noisy versions of low-rank quaternion matrices. Extensive experiments on both synthetic and real data further reveal the superiority of the proposed algorithm compared with other algorithms for getting low-rank approximation, in terms of both efficiency and accuracy.

翻译：低秩四元数矩阵逼近已成功应用于涉及信号处理和彩色图像处理的诸多领域。然而，由于需计算四元数奇异值分解(QSVD)，生成低秩四元数矩阵逼近的四元数模型成本有时相当可观，这限制了其在真实大规模数据中的应用。为解决此不足，本文提出了一种用于低秩逼近的高效四元数矩阵CUR(QMCUR)方法，该方法在彩色图像处理中实现了显著加速。我们首先探索了QMCUR逼近方法，该方法直接使用给定四元数矩阵的实际列和行，而非代价高昂的QSVD。此外，采用两种不同采样策略来选取上述列和行。随后，对带有噪声的低秩四元数矩阵的QMCUR逼近进行扰动分析。在合成数据和真实数据上的大量实验进一步揭示了所提算法在获取低秩逼近时，相较于其他算法在效率和精度方面的优越性。