New robust PCA for outliers and heavy sparse noises’ detection via affine transformation, the \(L_{\ast, w}\) and \(L_{2,1}\) norms, and spatial weight matrix in high-dimensional images: from the perspective of signal processing. (English) Zbl 1483.68436
Summary: In this paper, we propose a novel robust algorithm for image recovery via affine transformations, the weighted nuclear, \( L_{\ast, w}\), and the \(L_{2,1}\) norms. The new method considers the spatial weight matrix to account the correlated samples in the data, the \(L_{2,1}\) norm to tackle the dilemma of extreme values in the high-dimensional images, and the \(L_{\ast, w}\) norm newly added to alleviate the potential effects of outliers and heavy sparse noises, enabling the new approach to be more resilient to outliers and large variations in the high-dimensional images in signal processing. The determination of the parameters is involved, and the affine transformations are cast as a convex optimization problem. To mitigate the computational complexity, alternating iteratively reweighted direction method of multipliers (ADMM) method is utilized to derive a new set of recursive equations to update the optimization variables and the affine transformations iteratively in a round-robin manner. The new algorithm is superior to the state-of-the-art works in terms of accuracy on various public databases.
MSC:
| 68T45 | Machine vision and scene understanding |
| 62H25 | Factor analysis and principal components; correspondence analysis |
| 65K05 | Numerical mathematical programming methods |
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
References:
| [1] | Wright, J.; Yang, A. Y.; Ganesh, A.; Sastry, S. S.; Ma, Y., Robust face recognition via sparse representation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 31, 2, 210-227 (2008) |
| [2] | Yang, J.; Yu, K.; Gong, Y.; Huang, T., Linear spatial pyramid matching using sparse coding for image classification,, 2009 IEEE Conference on Computer Vision and Pattern Recognition, IEEE · doi:10.1109/cvpr.2009.5206757 |
| [3] | Barrile, V.; Candela, G.; Fotia, A., Point cloud segmentation using image processing techniques for structural analysis, The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, XLII-2, W11, 187-193 (2019) · doi:10.5194/isprs-archives-xlii-2-w11-187-2019 |
| [4] | Yan, Y.; Huang, L., Large-scale image processing research cloud, Proceedings of the International Conference on Cloud Computing |
| [5] | Agarwal, S.; Roth, D., Learning a sparse representation for object detection, Proceedings of the European Conference on Computer Vision, Springer · Zbl 1039.68590 · doi:10.1007/3-540-47979-1_8 |
| [6] | Xiang, X.; Tran, T. D., Linear disentangled representation learning for facial actions, IEEE Transactions on Circuits and Systems for Video Technology, 28, 12, 3539-3544 (2018) · doi:10.1109/tcsvt.2017.2771150 |
| [7] | Zhang, R.; Nie, F.; Li, X.; Wei, X., Feature selection with multi-view data: a survey, Information Fusion, 50, 158-167 (2019) · doi:10.1016/j.inffus.2018.11.019 |
| [8] | Wei, X.; Shen, H.; Li, Y.; Tang, X.; Wang, F.; Kleinsteuber, M.; Murphey, Y. L., Reconstructible nonlinear dimensionality reduction via joint dictionary learning, IEEE transactions on neural networks and learning systems, 30, 1, 175-189 (2018) |
| [9] | Lerman, G.; Maunu, T., An overview of robust subspace recovery, Proceedings of the IEEE, 106, 8, 1380-1410 (2018) · doi:10.1109/jproc.2018.2853141 |
| [10] | Huang, D.; Storer, M.; De la Torre, F.; Bischof, H., Supervised local subspace learning for continuous head pose estimation, CVPR 2011, IEEE · doi:10.1109/cvpr.2011.5995683 |
| [11] | Drouard, V.; Ba, S.; Evangelidis, G.; Deleforge, A.; Horaud, R., Head pose estimation via probabilistic high-dimensional regression,, Proceedings of the 2015 IEEE International Conference on Image Processing (ICIP), IEEE · doi:10.1109/icip.2015.7351683 |
| [12] | Wen, F.; Pei, L.; Yang, Y.; Yu, W.; Liu, P., Efficient and robust recovery of sparse signal and image using generalized nonconvex regularization, IEEE Transactions on Computational Imaging, 3, 4, 566-579 (2017) · doi:10.1109/tci.2017.2744626 |
| [13] | Candès, E. J.; Li, X.; Ma, Y.; Wright, J., Robust principal component analysis?, Journal of the ACM, 58, 3, 11 (2011) · Zbl 1327.62369 · doi:10.1145/1970392.1970395 |
| [14] | Ebadi, S.; Izquierdo, E., Approximated RPCA for fast and efficient recovery of corrupted and linearly correlated images and video frames, Proceedings of the International Conference on Systems, Signals and Image Processing (IWSSIP), IEEE · doi:10.1109/iwssip.2015.7314174 |
| [15] | Wright, J.; Ganesh, A.; Rao, S.; Peng, Y.; Ma, Y., Robust principal component analysis: exact recovery of corrupted low-rank matrices via convex optimization, Advances in Neural Information Processing Systems, 58, 2080-2088 (2009) |
| [16] | Wagner, A.; Wright, J.; Ganesh, A.; Zhou, Z.; Mobahi, H.; Ma, Y., Toward a practical face recognition system: robust alignment and illumination by sparse representation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 34, 2, 372-386 (2011) |
| [17] | Wei, X.; Shen, H.; Kleinsteuber, M., Trace quotient with sparsity priors for learning low dimensional image representations, IEEE Transactions on Pattern Analysis and Machine Intelligence, 42, 12, 3119-3135 (2019) |
| [18] | Oh, T.-H.; Tai, Y.-W.; Bazin, J.-C.; Kim, H.; Kweon, I. S., Partial sum minimization of singular values in robust PCA: algorithm and applications, IEEE Transactions on Pattern Analysis and Machine Intelligence, 38, 4, 744-758 (2016) · doi:10.1109/tpami.2015.2465956 |
| [19] | Lu, C.; Feng, J.; Chen, Y.; Liu, W.; Lin, Z.; Yan, S., Tensor robust principal component analysis with a new tensor nuclear norm, IEEE Transactions on Pattern Analysis and Machine Intelligence, 42, 4, 925-938 (2018) |
| [20] | Liu, Q.; Gu, Y.; So, H. C., Doa estimation in impulsive noise via low-rank matrix approximation and weakly convex optimization, IEEE Transactions on Aerospace and Electronic Systems, 55, 6, 3603-3616 (2019) · doi:10.1109/taes.2019.2909728 |
| [21] | Liu, Q.; Yang, C.; Gu, Y.; So, H. C., Robust sparse recovery via weakly convex optimization in impulsive noise, Signal Processing, 152, 84-89 (2018) · doi:10.1016/j.sigpro.2018.05.020 |
| [22] | Chen, X.; Han, Z.; Wang, Y.; Tang, Y.; Yu, H., Nonconvex plus quadratic penalized low-rank and sparse decomposition for noisy image alignment, Science China Information Sciences, 59, 5, 1-13 (2016) · doi:10.1007/s11432-015-5419-2 |
| [23] | Liu, Q.; Li, X.; Yang, J., Optimum co-design for image denoising between type-2 fuzzy identifier and matrix completion denoiser, IEEE Transactions on Fuzzy Systems (2020) · doi:10.1109/tfuzz.2020.3030498 |
| [24] | Shen, B.; Kamath, R. R.; Hahn Choo, Z. J. K., Robust tensor decomposition based background/foreground separation in noisy videos and its applications in additive manufacturing, TechRXiv (2021) |
| [25] | Cai, H.; Hamm, K.; Huang, L.; Needell, D., Robust cur decomposition: theory and imaging applications (2021), https://arxiv.org/2101.05231 |
| [26] | Moore, B. E.; Gao, C.; Nadakuditi, R. R., Panoramic robust PCA for foreground-background separation on noisy, free-motion camera video, IEEE Transactions on Computational Imaging, 5, 2, 195-211 (2019) · doi:10.1109/tci.2019.2891389 |
| [27] | De la Torre, F.; Black, M. J., Robust parameterized component analysis: theory and applications to 2D facial appearance models, Computer Vision and Image Understanding, 91, 1-2, 53-71 (2003) · doi:10.1016/s1077-3142(03)00076-6 |
| [28] | Nguyen, D. T., Long-rank Approximation and Sparse Recovery for Visual Data Reconstruction (2012), Baltimore, Maryland: The Johns Hopkins University, Baltimore, Maryland |
| [29] | Tao, M.; Yuan, X., Recovering low-rank and sparse components of matrices from incomplete and noisy observations, SIAM Journal on Optimization, 21, 1, 57-81 (2011) · Zbl 1218.90115 · doi:10.1137/100781894 |
| [30] | Vidal, R.; Ma, Y.; Sastry, S. S., Generalized Principal Component Analysis (2016), Berlin, Germany: Springer, Berlin, Germany, vol. 5 · Zbl 1349.62006 |
| [31] | Vidal, R.; Ma, Y.; Sastry, S. S., Robust principal component analysis, Interdisciplinary Applied Mathematics, 63-122 (2016), Springer · Zbl 1349.62006 · doi:10.1007/978-0-387-87811-9_3 |
| [32] | Baghaie, A.; D’souza, R. M.; Yu, Z., Sparse and low rank decomposition based batch image alignment for speckle reduction of retinal OCT images,, Proceedings of the 2015 IEEE 12th International Symposium on Biomedical Imaging (ISBI), IEEE · doi:10.1109/isbi.2015.7163855 |
| [33] | Likassa, H. T.; Fang, W.-H.; Chuang, Y.-A., Modified robust image alignment by sparse and low rank decomposition for highly linearly correlated data, Proceedings of the 2018 3rd International Conference on Intelligent Green Building and Smart Grid (IGBSG), IEEE · doi:10.1109/igbsg.2018.8393549 |
| [34] | Zheng, Q.; Wang, Y.; Heng, P.-A., Online robust image alignment via subspace learning from gradient orientations, Proceedings of the IEEE International Conference on Computer Vision (ICCV) · doi:10.1109/iccv.2017.195 |
| [35] | Xiao, S.; Tan, M.; Xu, D.; Dong, Z. Y., Robust kernel low-rank representation, IEEE transactions on neural networks and learning systems, 27, 11, 2268-2281 (2015) |
| [36] | Liu, G.; Lin, Z.; Yan, S.; Sun, J.; Yu, Y.; Ma, Y., Robust recovery of subspace structures by low-rank representation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35, 1, 171-184 (2012) |
| [37] | Obozinski, G. R.; Wainwright, M. J.; Jordan, M. I., High-dimensional support union recovery in multivariate regression, Advances in Neural Information Processing Systems, 21, 1217-1224 (2009) |
| [38] | Wei, X.; Li, Y.; Shen, H.; Chen, F.; Kleinsteuber, M.; Wang, Z., Dynamical textures modeling via joint video dictionary learning, IEEE Transactions on Image Processing, 26, 6, 2929-2943 (2017) · Zbl 1409.94653 · doi:10.1109/tip.2017.2691549 |
| [39] | Likassa, H. T., New robust principal component analysis for joint image alignment and recovery via affine transformations, frobenius and norms, International Journal of Mathematics and Mathematical Sciences, 2020 (2020) · Zbl 1486.90145 · doi:10.1155/2020/8136384 |
| [40] | Likassa, H. T.; Fang, W.-H.; Leu, J.-S., Robust image recovery via affine transformation and \(L_{2,1}\) norm, IEEE Access, 7, 125011-125021 (2019) · doi:10.1109/access.2019.2932470 |
| [41] | Lin, Z.; Chen, M.; Ma, Y., TThe augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices (2010), https://arxiv.org/1009.5055 |
| [42] | Liu, G.; Lin, Z.; Yu, Y., Robust subspace segmentation by low-rank representation, Proceedings of the 27th International Conference on Machine Learning (ICML-10) |
| [43] | Yigang Peng, Y.; Ganesh, A.; Wright, J.; Wenli Xu, W.; Yi Ma, Y., RASL: robust alignment by sparse and low-rank decomposition for linearly correlated images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 34, 11, 2233-2246 (2012) · doi:10.1109/tpami.2011.282 |
| [44] | Likassa, H. T.; Xian, W.; Tang, X., New robust regularized shrinkage regression for high-dimensional image recovery and alignment via affine transformation and tikhonov regularization, International Journal of Mathematics and Mathematical Sciences, 2020 (2020) · Zbl 1486.62211 · doi:10.1155/2020/1286909 |
| [45] | Wei, X.; Li, Y.; Shen, H.; Xiang, W.; Lu Murphey, Y., Joint learning sparsifying linear transformation for low-resolution image synthesis and recognition, Pattern Recognition, 66, 412-424 (2017) · doi:10.1016/j.patcog.2017.01.013 |
| [46] | Likassa, H. T.; Fang, W.-H., Robust regression for image alignment via subspace recovery techniques, Proceedings of the 2018 VII International Conference on Network, Communication and Computing · doi:10.1145/3301326.3301385 |
| [47] | Bouwmans, T.; Sobral, A.; Javed, S.; Jung, S. K.; Zahzah, E.-H., Decomposition into low-rank plus additive matrices for background/foreground separation: a review for a comparative evaluation with a large-scale dataset, Computer Science Review, 23, 1-71 (2017) · Zbl 1398.68572 · doi:10.1016/j.cosrev.2016.11.001 |
| [48] | Ding, C.; Zhou, D.; He, X.; Zha, H., R_1-PCA: rotational invariant L_1-norm principal component analysis for robust subspace factorization, Proceedings of the 23rd International Conference on Machine Learning (ICML) |
| [49] | Gu, S.; Zhang, L.; Zuo, W.; Feng, X., Weighted nuclear norm minimization with application to image denoising, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition · doi:10.1109/cvpr.2014.366 |
| [50] | Subbarao, R.; Meer, P., Beyond RANSAC: user independent robust regression, 2006 Conference on Computer Vision and Pattern Recognition Workshop (CVPRW’06), IEEE · doi:10.1109/cvprw.2006.43 |
| [51] | Boyd, S.; Parikh, N.; Chu, E.; Peleato, B.; Eckstein, J., Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends® in Machine Learning, 3, 1, 1-122 (2011) · Zbl 1229.90122 |
| [52] | Yang, J.; Zhang, Y., Alternating algorithms for 1-problems in compressive sensing (2010), Houston, TX: Rice University CAAM, Houston, TX, Tech. Rep. TR09-37 |
| [53] | Courrieu, P., Fast computation of moore-penrose inverse matrices (2008), https://arxiv.org/0804.4809 |
| [54] | LeCun, Y.; Bottou, L.; Bengio, Y.; Haffner, P., Gradient-based learning applied to document recognition, Proceedings of the IEEE, 86, 11, 2278-2324 (1998) · doi:10.1109/5.726791 |
| [55] | Huang, G. B.; Ramesh, M.; Berg, T.; Learned-Miller, E., Labeled Faces in the Wild: A Database for Studying Face Recognition in Unconstrained Environments (2007), Amherst, MA, USA: University of Massachusetts, Amherst, MA, USA |
| [56] | Hore, A.; Ziou, D., Image quality metrics: PSNR vs SSIM, 2010 20th International Conference on Pattern Recognition, IEEE · doi:10.1109/icpr.2010.579 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
