中文摘要：

目的 低秩矩阵恢复是通过最小化矩阵核范数来获得低秩解,然而待恢复低秩矩阵相关性低的要求往往会导致求解不稳定的情况.方法 针对该问题,研究一种基于变量分裂的低秩图像恢复去噪算法,引入待恢复矩阵的Fmbenius范数作为新正则项,与原有低秩矩阵的核范数组成联合正则化项,对问题进行凸松弛后,采用变量分裂的增广拉格朗日乘子法求解.结果 为考察方法的稳定性和去噪能力,选取了不同参数类型的加噪图像进行仿真,并结合恢复时间、信噪比、差错率等评价标准与现有低秩矩阵恢复算法进行对比.结论 实验结果表明增加Frobenius范数的低秩矩阵恢复模型在保持原有低秩稀疏恢复的前提下,具有良好的去噪性能,对相关性强的低秩图像恢复结果稳定性好,获得了更高的信噪比.

英文摘要：

Objective Low-rank matrix recovery is a hot topic in signal processing, artificial intelligence and optimiza- tion. Convex optimization based on low-rank matrix recovery problems coming from the compressive sensing technology, which is very popular these years for image processing, computer vision, text analysis, recommendation system, etc. Low- rank matrix recovery is achieved by minimizing the nuclear norm matrix to obtain a low rank solution, however, an unsta- ble solution can be obtained due to the requirements for the low correlation of a low rank matrix. Method A low rank image denoising algorithm is proposed based on variable splitting method. The method introduces a Frobenius norm of low rank ma- trices as a new regular item and it is also combined with the original low rank nuclear norm to optimize the image denois- ing. In order to solve the improved denoising model, an augmented Lagrange multiplier method based on variable splitting is used by using convex relaxation of sparse recovery methods. Result Finally, to verify the stability and denoising capability of the presented approach, images with different noise types and simulation parameters are generated and processed using the presented method and the results are compared with the existing low rank matrix algorithm. Performance analysis of re- covery time, signal-to-noise ratio, and error rate are evaluated at the same time. Conclusion The proposed method can yield superior performance compared to the traditional low rank model in terms of the test results. The experiments indicate that the improved models, while keeping the original low-rank sparse recovery, have good denoising performance and exce- llent stability on the strong correlation matrix and we can get a higher signal-to-noise ratio.