基于eLLp-L1-全变差正则化鲁棒主成分分析的运动目标检测
Moving Object Detection Based on eLLp-L1-Total Variation Regularized Robust Principal Component Analysis
摘要: 针对如何从受污染的高维数据中鲁棒地恢复内在的低维结构这一问题，本文综合了扩展LLp (eLLp)范数、L1范数和全变差(TV)正则化，提出了一种新的鲁棒主成分分析(RPCA)模型。针对基于核范数的传统RPCA对所有奇异值进行统一惩罚，忽略了图像和视频数据中固有的空间结构的缺陷，本文所提模型采用eLLp范数对秩函数提供更紧致的非凸近似，自适应地惩罚奇异值以区分显著和可忽略的分量。同时新模型中的TV正则化与L1范数分别促进了低秩分量的空间平滑性与误差分量的稀疏性。针对该模型，本文提出了一种基于交替方向乘子法的高效优化算法，同时讨论了算法的全局收敛结果。实际应用数据中的实验结果证明了所提方法在分解精度、视觉质量的优越性。
Abstract: Aiming at the problem of robustly recovering the intrinsic low-dimensional structure from contaminated high-dimensional data, this paper proposes a novel robust principal component analysis (RPCA) model by integrating the extended LLp (eLLp) norm, L1 norm, and total variation (TV) regularization. Traditional nuclear norm-based RPCA imposes uniform penalty on all singular values, overlooking the inherent spatial structure in image and video data. To address this limitation, the proposed model employs the eLLp norm to provide a tighter nonconvex approximation of the rank function, adaptively penalizing singular values to distinguish between significant and negligible components. Meanwhile, the TV regularization and L1 norm in the proposed model respectively promote the spatial smoothness of the low-rank component and the sparsity of the error component. An efficient optimization algorithm based on the alternating direction method of multipliers (ADMM) is developed to solve the proposed model, and the global convergence results of the algorithm are discussed. Experimental results on real-world data demonstrate the superiority of the proposed method in terms of decomposition accuracy and visual quality.
文章引用：祖铭辰. 基于eLLp-L1-全变差正则化鲁棒主成分分析的运动目标检测[J]. 计算机科学与应用, 2026, 16(3): 122-135. https://doi.org/10.12677/csa.2026.163092

参考文献

[1] Fan, J., Han, F. and Liu, H. (2014) Challenges of Big Data Analysis. National Science Review, 1, 293-314. [Google Scholar] [CrossRef] 
[2] Hastie, T., Tibshirani, R. and Friedman, J. (2009) An Introduction to Statistical Learning. Springer.
[3] Udell, M. and Townsend, A. (2019) Why Are Big Data Matrices Approximately Low Rank? SIAM Journal on Mathe-matics of Data Science, 1, 144-160. [Google Scholar] [CrossRef] 
[4] Koren, Y., Bell, R. and Volinsky, C. (2009) Matrix Factorization Techniques for Recommender Systems. Computer, 42, 30-37. [Google Scholar] [CrossRef] 
[5] Vaswani, N. and Narayanamurthy, P. (2018) Static and Dynamic Robust PCA and Matrix Completion: A Review. Proceedings of the IEEE, 106, 1359-1379. [Google Scholar] [CrossRef] 
[6] Wright, J., Yang, A.Y., Ganesh, A., Sastry, S.S. and Ma, Y. (2009) Robust Face Recognition via Sparse Representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31, 210-227. [Google Scholar] [CrossRef] 
[7] Chandola, V., Banerjee, A. and Kumar, V. (2009) Anomaly Detection: A Survey. ACM Computing Surveys, 41, 1-58. [Google Scholar] [CrossRef] 
[8] Jolliffe, I. (2011) Principal Component Analysis. In: Lovric, M., Ed., Internation-al Encyclopedia of Statistical Science, Springer, 1094-1096. [Google Scholar] [CrossRef] 
[9] Zhou, X., Yang, C., Zhao, H. and Yu, W. (2014) Low-Rank Modeling and Its Applications in Image Analysis. ACM Computing Surveys, 47, 1-33. [Google Scholar] [CrossRef] 
[10] Vaswani, N., Bouwmans, T., Javed, S. and Narayanamurthy, P. (2018) Robust Subspace Learning: Robust PCA, Robust Subspace Tracking, and Robust Subspace Recovery. IEEE Signal Processing Magazine, 35, 32-55. [Google Scholar] [CrossRef] 
[11] Candès, E.J., Li, X., Ma, Y. and Wright, J. (2011) Robust Principal Component Analysis? Journal of the ACM, 58, 1-37. [Google Scholar] [CrossRef] 
[12] Lin, Z., Chen, M. and Ma, Y. (2010) The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices. arXiv: 1009.5055.
[13] Neal, P., Eric, C., Borja, P., et al. (2010) Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers. Foundations and Trends® in Machine Learning, 3, 1-122. [Google Scholar] [CrossRef] 
[14] Zhou, Z., Li, X., Wright, J., Candes, E. and Ma, Y. (2010) Stable Principal Component Pursuit. 2010 IEEE International Symposium on Information Theory, Austin, 13-18 June 2010, 1518-1522. [Google Scholar] [CrossRef] 
[15] Peng, Y.G., Ganesh, A., Wright, J., Xu, W.L. and Ma, Y. (2012) RASL: Robust Alignment by Sparse and Low-Rank Decomposition for Linearly Correlated Images. IEEE Transactions on Pattern Analysis and Ma-chine Intelligence, 34, 2233-2246. [Google Scholar] [CrossRef] 
[16] Lu, C., Feng, J., Chen, Y., Liu, W., Lin, Z. and Yan, S. (2020) Tensor Robust Principal Component Analysis with a New Tensor Nuclear Norm. IEEE Transactions on Pattern Analy-sis and Machine Intelligence, 42, 925-938. [Google Scholar] [CrossRef] 
[17] He, R., Zheng, W.S. and Hu, B.G. (2011) Maximum Correntropy Criterion for Robust Face Recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 33, 1561-1576. [Google Scholar] [CrossRef] 
[18] Hu, Y., Zhang, D., Ye, J., Li, X. and He, X. (2013) Fast and Accurate Matrix Com-pletion via Truncated Nuclear Norm Regularization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35, 2117-2130. [Google Scholar] [CrossRef] 
[19] Gu, S., Xie, Q., Meng, D., Zuo, W., Feng, X. and Zhang, L. (2016) Weighted Nu-clear Norm Minimization and Its Applications to Low Level Vision. International Journal of Computer Vision, 121, 183-208. [Google Scholar] [CrossRef] 
[20] Lu, C., Tang, J., Yan, S. and Lin, Z. (2014) Generalized Nonconvex Nonsmooth Low-Rank Minimization. 2014 IEEE Conference on Computer Vision and Pattern Recognition, Columbus, 23-28 June 2014, 4130-4137. [Google Scholar] [CrossRef] 
[21] Fazel, M., Hindi, H. and Boyd, S.P. (2003) Log-Det Heuristic for Matrix Rank Min-imization with Applications to Hankel and Euclidean Distance Matrices. Proceedings of the 2003 American Control Conference, 2003, Denver, 4-6 June 2003, 2156-2162. [Google Scholar] [CrossRef] 
[22] Zhang, C. (2010) Nearly Unbiased Variable Selection under Minimax Concave Penalty. The Annals of Statistics, 38, 894-942. [Google Scholar] [CrossRef] 
[23] Wang, L., Wu, Y. and Li, R. (2012) Quantile Regression for Analyzing Heterogeneity in Ultra-High Dimension. Journal of the American Sta-tistical Association, 107, 214-222. [Google Scholar] [CrossRef] 
[24] Keshavarzian, R. and Aghagolzadeh, A. (2023) Low Rank and Sparse Decomposition Based on Extended LLP Norm. Multimedia Tools and Applications, 83, 26107-26130. [Google Scholar] [CrossRef] 
[25] Bouwmans, T., Javed, S., Zhang, H., Lin, Z. and Otazo, R. (2018) On the Ap-plications of Robust PCA in Image and Video Processing. Proceedings of the IEEE, 106, 1427-1457. [Google Scholar] [CrossRef] 
[26] Rudin, L.I., Osher, S. and Fatemi, E. (1992) Nonlinear Total Variation Based Noise Removal Algorithms. Physica D: Nonlinear Phenomena, 60, 259-268. [Google Scholar] [CrossRef] 
[27] He, W., Zhang, H., Zhang, L. and Shen, H. (2016) To-tal-Variation-Regularized Low-Rank Matrix Factorization for Hyperspectral Image Restoration. IEEE Transactions on Geoscience and Remote Sensing, 54, 178-188. [Google Scholar] [CrossRef] 
[28] Dabov, K., Foi, A., Katkovnik, V. and Egiazarian, K. (2007) Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering. IEEE Transactions on Image Processing, 16, 2080-2095. [Google Scholar] [CrossRef] 
[29] Sun, Y., Babu, P. and Palomar, D.P. (2017) Majorization-minimization Algorithms in Signal Processing, Communications, and Machine Learning. IEEE Transactions on Signal Processing, 65, 794-816. [Google Scholar] [CrossRef] 
[30] Beck, A. and Teboulle, M. (2009) A Fast Iterative Shrinkage-Thresholding Algo-rithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2, 183-202. [Google Scholar] [CrossRef] 
[31] Keshavarzian, R., Aghagolzadeh, A. and Rezaii, T.Y. (2019) LLP Norm Regularization Based Group Sparse Representation for Image Compressed Sensing Recovery. Signal Processing: Image Communication, 78, 477-493. [Google Scholar] [CrossRef] 
[32] Dong, W., Shi, G., Li, X., Ma, Y. and Huang, F. (2014) Compressive Sensing via Nonlocal Low-Rank Regularization. IEEE Transactions on Image Processing, 23, 3618-3632. [Google Scholar] [CrossRef] 
[33] Chen, K., Dong, H. and Chan, K. (2013) Reduced Rank Regression via Adaptive Nuclear Norm Penalization. Biometrika, 100, 901-920. [Google Scholar] [CrossRef] 
[34] Wang, Y., Yin, W. and Zeng, J. (2018) Global Convergence of ADMM in Nonconvex Nonsmooth Optimization. Journal of Scientific Computing, 78, 29-63. [Google Scholar] [CrossRef] 
[35] Hong, M., Luo, Z. and Razaviyayn, M. (2016) Convergence Analysis of Alter-nating Direction Method of Multipliers for a Family of Nonconvex Problems. SIAM Journal on Optimization, 26, 337-364. [Google Scholar] [CrossRef] 
[36] Shijila, B., Tom, A.J. and George, S.N. (2018) Moving Object Detection by Low Rank Approximation Andl1-Tv Regularization on RPCA Framework. Journal of Visual Communication and Image Representation, 56, 188-200. [Google Scholar] [CrossRef] 
[37] Kang, Z., Peng, C. and Cheng, Q. (2015) Robust PCA via Nonconvex Rank Approximation. 2015 IEEE International Conference on Data Mining, Atlantic City, 14-17 November 2015, 211-220. [Google Scholar] [CrossRef]