A novel robust principal component analysis method for image and video processing. (English) Zbl 1389.62096

Summary: The research on the robust principal component analysis has been attracting much attention recently. Generally, the model assumes sparse noise and characterizes the error term by the \(\ell_1\)-norm. However, the sparse noise has clustering effect in practice so using a certain \(\ell_p\)-norm simply is not appropriate for modeling. In this paper, we propose a novel method based on sparse Bayesian learning principles and Markov random fields. The method is proved to be very effective for low-rank matrix recovery and contiguous outliers detection, by enforcing the low-rank constraint in a matrix factorization formulation and incorporating the contiguity prior as a sparsity constraint. The experiments on both synthetic data and some practical computer vision applications show that the novel method proposed in this paper is competitive when compared with other state-of-the-art methods.


62H25 Factor analysis and principal components; correspondence analysis
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
68Q87 Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)
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[1] Babacan, S. D.; Luessi, M.; Molina, R.; Katsaggelos, A. K., Sparse Bayesian methods for low-rank matrix estimation, IEEE Trans. Signal Process, 60, 3964-3977, (2012) · Zbl 1393.94670
[2] Bishop, C. M., Pattern recognition and machine learning, (2006), New York · Zbl 1107.68072
[3] Boykov, Y.; Veksler, O.; Zabih, R., Fast approximate energy minimization via graph cuts, IEEE Trans. Pattern Anal. Mach. Intell., 23, 1222-1239, (2010)
[4] E. J. Candès, X. Li, Y. Ma, J. Wright: Robust principal component analysis? J. ACM 58 (2011), Article No. 11, 37 pages. · Zbl 1327.62369
[5] Candès, E. J.; Recht, B., Exact matrix completion via convex optimization, Found. Comput. Math., 9, 717-772, (2009) · Zbl 1219.90124
[6] Candès, E. J.; Tao, T., The power of convex relaxation: near-optimal matrix completion, IEEE Trans. Inf. Theory, 56, 2053-2080, (2010) · Zbl 1366.15021
[7] Torre, F.; Black, M. J., A framework for robust subspace learning, Int. J. Comput. Vis., 54, 117-142, (2003) · Zbl 1076.68058
[8] Ding, X.; He, L.; Carin, L., Bayesian robust principal component analysis, IEEE Trans. Image Process., 20, 3419-3430, (2011) · Zbl 1381.62144
[9] Ding, C.; Zhou, D.; He, X.; Zha, H., \(R\)_{1}-PCA: rotational invariant \(L\)_{1}-norm principal component analysis for robust subspace factorization, 281-288, (2006), New York
[10] Hansen, P. C., Rank-deficient and discrete ill-posed problems, No. 4, (1998), Philadelphia
[11] Jolliffe, I. T., Principal component analysis, (2002), New York · Zbl 1011.62064
[12] Kolmogorov, V.; Zabin, R., What energy functions can be minimized via graph cuts?, IEEE Trans. Pattern Anal. Mach. Intell., 26, 147-159, (2004)
[13] Kwak, N., Principal component analysis based on L1-norm maximization, IEEE Trans. Pattern Anal. Mach. Intell., 30, 1672-1680, (2008)
[14] Li, S. Z., Markov random field modeling in image analysis, (2009), London · Zbl 1183.68691
[15] Z. Lin, M. Chen, Y. Ma: The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices. ArXiv preprint arXiv:1009. 5055, 2010. · Zbl 1076.68058
[16] Z. Lin, A. Ganesh, J. Wright, L. Wu, M. Chen, Y. Ma: Fast convex optimization algorithms for exact recovery of a corrupted low-rank matrix. Computational Advances in Multi-Sensor Adaptive Processing. Aruba, Dutch Antilles, Proceedings. IEEE, 2009. · Zbl 1198.90321
[17] G. Liu, Z. Lin, Y. Yu: Robust subspace segmentation by low-rank representation. Proc. 27th Int. Conf. on Machine Learning, Haifa. Proceedings Omni Press., 2010, pp. 663-670. · Zbl 1365.62228
[18] Mazumder, R.; Hastie, T.; Tibshirani, R., Spectral regularization algorithms for learning large incomplete matrices, J. Mach. Learn. Res., 11, 2287-2322, (2010) · Zbl 1242.68237
[19] Peng, Y.; Ganesh, A.; Wright, J.; Xu, W.; Ma, Y., RASL: robust alignment by sparse and low-rank decomposition for linearly correlated images, IEEE Trans. Pattern Anal. Mach. Intell., 34, 2233-2246, (2012)
[20] Recht, B.; Fazel, M.; Parrilo, P. A., Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev., 52, 471-501, (2010) · Zbl 1198.90321
[21] N. Wang, D. Y. Yeung: Bayesian robust matrix factorization for image and video processing. Computer Vision, 2013 IEEE International Conference, Sydney. Proceedings. IEEE, 2013, pp. 1785-1792. · Zbl 1011.62064
[22] J. Wright, A. Ganesh, S. Rao, Y. Peng, Y. Ma: Robust principal component analysis: Exact recovery of corrupted low-rank matrices via convex optimization. Advances in Neural Information Processing Systems 22, Vancouver. Proceedings. Curran Associates, 2009, pp. 2080-2088.
[23] Xu, H.; Caramanis, C.; Sanghavi, S., Robust PCA via outlier pursuit, IEEE Trans. Inform. Theory, 58, 3047-3064, (2012) · Zbl 1365.62228
[24] Q. Zhao, D. Meng, Z. Xu, W. Zuo, L. Zhang: Robust principal component analysis with complex noise. Proc. 31st Int. Conf. on Machine Learning, Beijing. Proceedings. J. Mach. Learn Res., 2014, pp. 55-63. · Zbl 1219.90124
[25] Z. Zhou, X. Li, J. Wright, E. J. Candès, Y. Ma: Stable principal component pursuit. 2010 IEEE International Symposium on Information Theory Proceedings, Austin. Proceedings. IEEE, 2010, pp. 1518-1522.
[26] Zhou, X.; Yang, C.; Yu, W., Moving object detection by detecting contiguous outliers in the low-rank representation, IEEE Trans. Pattern Anal. Mach. Intell., 35, 597-610, (2013)
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