中文摘要：

近年来,低秩矩阵重建在机器学习、图像处理、计算机视觉与生物信息学等众多科学与工程应用领域中,迅速发展为一个新的研究热点,其主要涉及矩阵填充与稳健主成分分析2大问题,即分别从精确且不完全的采样矩阵元与从大误差矩阵元的分布较为稀疏的观测矩阵中恢复出原始低秩矩阵。鉴此,本文定义了稳健矩阵填充,即从非完全且存在稀疏误差的采样矩阵元中精确恢复出原始低秩矩阵,通过最小化核范数与l1-范数的组合构建了相应的凸优化模型,并提出了一种新颖的增广分部拉格朗日乘数法来求解此类最优化问题。通过将其应用于混合高斯与椒盐噪声去除的问题中表明,此算法对具有规则纹理及相似结构内容等低秩特征的影像中混合噪声的去除效果较好,其能同时去除影像中的椒盐噪声与高斯噪声,且有效保留影像中的纹理细节等信息;当影像中椒盐噪声密度较高而高斯噪声相对较小时,其去噪性能更佳。

英文摘要：

This paper studies the problem of the restoration of images corrupted by mixed Gaussian-impulse noise. In recent years, low-rank matrix reconstruction has become a research hotspot in many scientific and engineering domains such as machine learning, image processing, computer vision and bioinformatics, which mainly involves the problems of matrix completion and robust principal component analysis. The two problems namely focus on recovering a low-rank matrix from an incomplete but accurate sampling subset of its entries, or from an observed data matrix with an unknown fraction of its entries being arbitrarily corrupted, respectively. Inspired by these ideas, the problem of recovering a low-rank matrix from an incomplete sampling subset of its entries with an unknown fraction of the samplings contaminated by arbitrary errors was considered, which was defined as a problem of matrix completion from corrupted samplings and modeled as a convex optimization problem that minimizes a combination of the nuclear norm and the/l-norm in this paper. Meanwhile, a novel and effective algorithm called augmented subsection Lagrange multipliers was put forward to exactly solve the problem. For the mixed Gaussian-impulse noise removal, we regard it as the problem of matrix completion from corrupted sam- plings, and restore the noisy images following by an impulse-detecting procedure. Compared with some existing methods for mixed noise removal, the recovery quality of our method is dominant when the images possess lowrank features such as geometrically regular textures and similar structural contents. Especially when the density of impulse noise is relatively high and the variance of Gaussian noise is small, our method can outperform the traditional methods significantly not only in the simultaneous removal of Gaussian noise and impulse noise, and in the restoration of low-rank image matrix, but also in the preservation of textures and details of the image.