目的 针对因采集的人脸图像样本受到污染而严重干扰人脸识别及训练样本较少（小样本）时会由于错误的稀疏系数导致性能急剧下降从而影响人脸识别的问题，提出了一种基于判别性非凸低秩矩阵分解的叠加线性稀疏表示算法。方法 首先由γ范数取代传统核范数，克服了传统低秩矩阵分解方法求解核范数时因矩阵奇异值倍数缩放导致的识别误差问题；然后引入结构不相干判别项，以增加不同类低秩字典间的非相干性，达到抑制类内变化和去除类间相关性的目的；最后利用叠加线性稀疏表示方法完成分类。结果 所提算法在AR人脸库中的识别率达到了98.67±0.57%，高于SRC（sparse representation-based classification）、ESRC（extended SRC）、RPCA（robust principal component analysis）+SRC、LRSI（low rank matrix decomposition with structural incoherence）、SLRC（superposed linear representation based classification）-l₁等算法；同时，遮挡实验表明，算法对遮挡图像具有更好的鲁棒性，在不同遮挡比例下，相比其他算法均有更高的识别率。在CMU PIE人脸库中，对无遮挡图像添加0、10%、20%、30%、40%的椒盐噪声，算法识别率分别达到90.1%、85.5%、77.8%、65.3%和46.1%，均高于其他算法。结论 不同人脸库、不同比例遮挡和噪声的实验结果表明，所提算法针对人脸遮挡、表情和光照等噪声因素依然保持较高的识别率，鲁棒性更好。

Objective Face recognition has become one of the most popular biometric recognition methods because of rich facial information and wide application prospects. However, the quality of images taken by the equipment is often affected in the real environment. Various facial expressions, gestures, and illumination conditions will affect the quality of face images, resulting in occlusion, translation, and scale errors in normalized face images, thereby reducing the robustness and recognition accuracy of face recognition algorithms. Among many known algorithms, the sparse representation-based classification (SRC) algorithm has achieved good face recognition performance. The algorithm is robust to noise and partial occlusion. However, face recognition in cases of facial expression change, posture change, and small sample size remains a challenge. On the one hand, the SRC can be used successfully in face recognition when the training samples are sufficient, where a testing sample can be represented by a linear combination of the images of the same person in the database. On the other hand, the SRC will divide samples into the wrong classes due to misleading coefficients on the under-sampled database. Therefore, studying how to obtain better recognition results under polluted and small samples remains important. On the basis of this situation, this work aims to study a face recognition algorithm with SRC on an uncontrolled and under-sampled database. Method This study proposes a superposition linear sparse representation face recognition algorithm based on discriminant non-convex low-rank matrix decomposition considering the sparsity of the SRC method and the low-rank matrix decomposition because the low-rank matrix decomposition has a good effect on removing sample noise. A dictionary that can eliminate interclass correlation is obtained by decomposing the training samples twice. This dictionary is used to classify and recognize by reconstructing the sparse residual model. The proposed algorithm efficiently eliminates the errors caused by occlusion and other unavoidable factors. We utilize non-convex rank approximation (norm) to replace nuclear norm due to two major limitations of robust principal component analysis (RPCA):the lack of structural incoherence and the tendency to shrink all the singular values equally. Non-convex rank approximation overcomes the problem of the singular value of the matrix being scaled to the same multiple when solving the kernel norm by using traditional RPCA method, which may lead to errors in the recognition results. We add to the theory of structural irrelevance in low-rank decomposition to minimize the Frobenius norm among all kinds of low-rank dictionaries and the between-class scatter. In addition, this method increases the incoherence among the low-rank dictionaries, thus improving the discrimination ability of low-rank matrices. After obtaining the low-rank matrix, the classification is completed by superposed linear sparse representation classification (SLRC). We divide the low-rank matrix into the prototype dictionary and the variation dictionary according to SLRC. Then, the two dictionaries are combined into a training dictionary in SRC. The homotopy method is used to obtain the sparse coefficients of l₁ norm. Furthermore, the dictionary is classified by reconstructing the sparse-minimizing residual model. This study eliminates the interference of intraclass/interclass correlation and even on the under-sampled database. Result This study selects the AR and CMU PIE databases for experiments. In the AR database, the recognition rate of our algorithm is 98.67 ±0.57% in 10 experiments, which is better than that of SRC, extended SRC, RPCA+SRC, low-rank matrix decomposition with structural incoherence (LRSI), and superposed linear representation-based classification (SLRC-l₁), among others. We choose different proportions from 0 to 3/7 of occluded pictures of people wearing a scarf or sunglasses. This condition means that the number of sunglasses or scarf increased from zero to three, and the total number of training images per class is seven. Our algorithm has better robustness to occluded images and has a higher recognition rate than other algorithms. The difference in the recognition rate of the algorithms decreases gradually as the number of occlusion training samples increases because the low-rank matrix decomposed by RPCA lacks discriminant information, even though the recognition rate of RPCA + SRC is better than that of SRC in all test samples. The proposed algorithm will not be affected by the presence of structural irrelevant terms. In addition, when the proportion of occluded pictures is increased in the training samples, the recognition performance of each algorithm is gradually improved, because the SRC part of the algorithm is more sensitive to the information of occluded pictures. For the CMU PIE database, we add salt-and-pepper noise from 0 to 40% in every image. The recognition rate of our algorithm reaches 90.1%, 85.5%, 77.8%, 65.3%, and 46.1%, and is the highest among all the compared algorithms. In different methods, the recognition rate of the SRC-based method decreases significantly with the increase in the percentage of damaged pixels. The RPCA-based method performs better in noise elimination. Therefore, adding low-rank decomposition to the algorithm helps improve the recognition rate. Conclusion The proposed algorithm has a high recognition rate in different face databases, especially in the case of occlusion and noise pollution. In summary, discriminant non-convex low-rank matrix decomposition is introduced into superposed linear sparse representation, which improves the robustness and efficiency of recognition considerably because of improved recognition accuracy. Thus, the algorithm has good application value in practice. The proposed algorithm is only for static images; thus, how to combine static face recognition with dynamic video to effectively realize video-based face recognition needs further research.