Abstract
For a given data, robust principal component analysis (RPCA) aims to exactly recover the low-rank and sparse components from it. To date, as the convex relaxations of tensor rank, a number of tensor nuclear norms have been defined and applied to approximate the tensor rank because of their convexity. However, those relaxations may make the solution seriously deviate from the original solution for real-world data recovery. In this paper, we define the tensor Schatten p-norm based on tensor train rank and propose a new model for tensor robust principal component analysis (named \(\hbox {TT}S_{p}\)). We solve the proposed model iteratively by using the ADMM algorithm. In addition, a tensor augmentation tool called ket augmentation is introduced to convert lower-order tensors to higher-order tensors to exploit the low-TT-rank structure. We report higher PSNR and SSIM values in numerical experiments to image recovery problems which demonstrate the superiority of our method. Further experiments on real data also illustrate the effectiveness of the proposed method.
Similar content being viewed by others
Notes
References
Cao, L., Yang, X., Guo, X.: Total variation regularized RPCA for irregularly moving object detection under dynamic background. IEEE Trans. Cybern. 46(4), 1014–1027 (2015)
Cao, W., Wang, Y., Sun, J., Meng, D., Yang, C., Cichocki, A., Xu, Z.: Total variation regularized tensor RPCA for background subtraction from compressive measurements. IEEE Trans. Image Process. 25(9), 4075–4090 (2016)
Yang, J.-H., Zhao, X.-L., Ji, T.-Y., Ma, T.-H., Huang, T.-Z.: Low-rank tensor train for tensor robust principal component analysis. Appl. Math. Comput. 367, 124783 (2020)
Sun, W., Yang, G., Peng, J., Du, Q.: Lateral-slice sparse tensor robust principal component analysis for hyperspectral image classification. IEEE Geosci. Remote Sens. Lett. 17(1), 107–111 (2019)
Li, X., Shen, H., Zhang, L., Zhang, H., Yuan, Q., Yang, G.: Recovering quantitative remote sensing products contaminated by thick clouds and shadows using multitemporal dictionary learning. IEEE Trans. Geosci. Remote Sens. 52(11), 7086–7098 (2014). https://doi.org/10.1109/TGRS.2014.2307354
Lin, C., Wang, Y., Wang, T., Ni, D.: Segmentation and recovery of pathological mr brain images using transformed low-rank and structured sparse decomposition. In: 2019 IEEE 16th International Symposium on Biomedical Imaging (ISBI 2019), pp. 1878–1881 (2019). https://doi.org/10.1109/ISBI.2019.8759441
Chen, Y., He, W., Yokoya, N., Huang, T.-Z.: Hyperspectral image restoration using weighted group sparsity-regularized low-rank tensor decomposition. IEEE Trans. Cybern. 50(8), 3556–3570 (2020). https://doi.org/10.1109/TCYB.2019.2936042
Ladas, N., Kaimakis, P., Chrysanthou, Y.: Background segmentation in multicolored illumination environments. Vis. Comput. 37(8), 2221–2233 (2021)
Gao, Z., Cheong, L.-F., Wang, Y.-X.: Block-sparse RPCA for salient motion detection. IEEE Trans. Pattern Anal. Mach. Intell. 36(10), 1975–1987 (2014). https://doi.org/10.1109/TPAMI.2014.2314663
Xie, Q., Zhao, Q., Meng, D., Xu, Z.: Kronecker-basis-representation based tensor sparsity and its applications to tensor recovery. IEEE Trans. Pattern Anal. Mach. Intell. 40(8), 1888–1902 (2018). https://doi.org/10.1109/TPAMI.2017.2734888
Tu, Z., Guo, Z., Xie, W., Yan, M., Veltkamp, R.C., Li, B., Yuan, J.: Fusing disparate object signatures for salient object detection in video. Pattern Recogn. 72, 285–299 (2017). https://doi.org/10.1016/j.patcog.2017.07.028
Abdi, H., Williams, L.J.: Principal component analysis. Wiley interdiscip. Rev. Comput. Stat. 2(4), 433–459 (2010)
Candès, E.J., Li, X., Ma, Y., Wright, J.: Robust principal component analysis? J. ACM (JACM) 58(3), 1–37 (2011)
Wright, J., Ganesh, A., Rao, S.R., Peng, Y., Ma, Y.: Robust principal component analysis: Exact recovery of corrupted low-rank matrices via convex optimization. In: NIPS, vol. 58, pp. 289–298 (2009)
Croux, C., Haesbroeck, G.: Principal component analysis based on robust estimators of the covariance or correlation matrix: influence functions and efficiencies. Biometrika 87(3), 603–618 (2000)
Tang, G., Nehorai, A.: Robust principal component analysis based on low-rank and block-sparse matrix decomposition. In: 2011 45th Annual Conference on Information Sciences and Systems, pp. 1–5 (2011). https://doi.org/10.1109/CISS.2011.5766144
Zhang, H., Cai, J., Cheng, L., Zhu, J.: Strongly convex programming for exact matrix completion and robust principal component analysis. arXiv preprint arXiv:1112.3946 (2011). https://doi.org/10.48550/arXiv.1112.3946
Bhardwaj, A., Raman, S.: Robust PCA-based solution to image composition using augmented Lagrange multiplier (ALM). Vis. Comput. 32(5), 591–600 (2016)
Giraldo-Zuluaga, J.-H., Salazar, A., Gomez, A., Diaz-Pulido, A.: Camera-trap images segmentation using multi-layer robust principal component analysis. Vis. Comput. 35(3), 335–347 (2019)
Xue, Z., Dong, J., Zhao, Y., Liu, C., Chellali, R.: Low-rank and sparse matrix decomposition via the truncated nuclear norm and a sparse regularizer. Vis. Comput. 35(2), 1549–1566 (2019)
Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)
Yang, J.-H., Zhao, X.-L., Ma, T.-H., Ding, M., Huang, T.-Z.: Tensor train rank minimization with hybrid smoothness regularization for visual data recovery. Appl. Math. Model. 81, 711–726 (2020)
Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)
Gandy, S., Recht, B., Yamada, I.: Tensor completion and low-n-rank tensor recovery via convex optimization. Inverse Prob. 27(2), 025010 (2011). https://doi.org/10.1088/0266-5611/27/2/025010
Kressner, D., Steinlechner, M., Vandereycken, B.: Low-rank tensor completion by riemannian optimization. BIT Numer. Math. 54(2), 447–468 (2014)
Xu, Y., Hao, R., Yin, W., Su, Z.: Parallel matrix factorization for low-rank tensor completion. arXiv preprint arXiv:1312.1254 (2013)
Liu, J., Musialski, P., Wonka, P., Ye, J.: Tensor completion for estimating missing values in visual data. IEEE Trans. Pattern Anal. Mach. Intell. 35(1), 208–220 (2012)
Huang, B., Mu, C., Goldfarb, D., Wright, J.: Provable models for robust low-rank tensor completion. Pac. J. Optim. 11(2), 339–364 (2015)
Gao, S., Fan, Q.: Robust Schatten p-norm based approach for tensor completion. J. Sci. Comput. 82(1), 1–23 (2020)
Cao, W., Wang, Y., Sun, J., Meng, D., Yang, C., Cichocki, A., Xu, Z.: Total variation regularized tensor RPCA for background subtraction from compressive measurements. IEEE Trans. Image Process. 25(9), 4075–4090 (2016)
Bengua, J.A., Phien, H.N., Tuan, H.D., Do, M.N.: Efficient tensor completion for color image and video recovery: low-rank tensor train. IEEE Trans. Image Process. 26(5), 2466–2479 (2017)
Kilmer, M.E., Braman, K., Hao, N., Hoover, R.C.: Third-order tensors as operators on matrices: a theoretical and computational framework with applications in imaging. SIAM J. Matrix Anal. Appl. 34(1), 148–172 (2013)
Lu, C., Feng, J., Chen, Y., Liu, W., Lin, Z., Yan, S.: Tensor robust principal component analysis: exact recovery of corrupted low-rank tensors via convex optimization. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5249–5257 (2016)
Zhang, Z., Ely, G., Aeron, S., Hao, N., Kilmer, M.: Novel methods for multilinear data completion and de-noising based on tensor-SVD. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3842–3849 (2014)
Semerci, O., Hao, N., Kilmer, M.E., Miller, E.L.: Tensor-based formulation and nuclear norm regularization for multienergy computed tomography. IEEE Trans. Image Process. 23(4), 1678–1693 (2014)
Lu, C., Feng, J., Chen, Y., Liu, W., Lin, Z., Yan, S.: Tensor robust principal component analysis with a new tensor nuclear norm. IEEE Trans. Pattern Anal. Mach. Intell. 42(4), 925–938 (2019)
Gao, Q., Zhang, P., Xia, W., Xie, D., Gao, X., Tao, D.: Enhanced tensor RPCA and its application. IEEE Trans. Pattern Anal. Mach. Intell. 43(6), 2133–2140 (2021). https://doi.org/10.1109/TPAMI.2020.3017672
Oseledets, I.V.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)
Lu, C., Tang, J., Yan, S., Lin, Z.: Nonconvex nonsmooth low rank minimization via iteratively reweighted nuclear norm. IEEE Trans. Image Process. 25(2), 829–839 (2016). https://doi.org/10.1109/TIP.2015.2511584
Parekh, A., Selesnick, I.W.: Enhanced low-rank matrix approximation. IEEE Signal Process. Lett. 23(4), 493–497 (2016). https://doi.org/10.1109/LSP.2016.2535227
Larsson, V., Olsson, C.: Convex low rank approximation. Int. J. Comput. Vision 120(2), 194–214 (2016)
Latorre, J.I.: Image compression and entanglement. arXiv preprint arXiv:quant-ph/0510031 (2005)
Luo, Q., Liu, B., Zhang, Y., Han, Z., Tang, Y.: Low-rank decomposition on transformed feature maps domain for image denoising. Vis. Comput. 37(7), 1899–1915 (2021)
Nie, F., Wang, H., Cai, X., Huang, H., Ding, C.: Robust matrix completion via joint Schatten p-norm and LP-norm minimization. In: 2012 IEEE 12th International Conference on Data Mining, pp. 566–574. IEEE (2012)
Peng, C., Chen, Y., Kang, Z., Chen, C., Cheng, Q.: Robust principal component analysis: a factorization-based approach with linear complexity. Inf. Sci. 513, 581–599 (2020). https://doi.org/10.1016/j.ins.2019.09.074
Healey, G.E., Kondepudy, R.: Radiometric CCD camera calibration and noise estimation. IEEE Trans. Pattern Anal. Mach. Intell. 16(3), 267–276 (1994)
Bovik, A.C.: Handbook of Image and Video Processing, pp. 90–91. Academic Press, Burlington (2010)
Ji, H., Huang, S., Shen, Z., Xu, Y.: Robust video restoration by joint sparse and low rank matrix approximation. SIAM J. Imaging Sci. 4(4), 1122–1142 (2011)
Martin, D.R., Fowlkes, C.C., Tal, D., Malik, J.: A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics. In: Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001, vol. 2, pp. 416–423 (2001). IEEE
Jiang, T.-X., Huang, T.-Z., Zhao, X.-L., Deng, L.-J., Wang, Y.: Fastderain: a novel video rain streak removal method using directional gradient priors. IEEE Trans. Image Process. 28(4), 2089–2102 (2019). https://doi.org/10.1109/TIP.2018.2880512
Maddalena, L., Petrosino, A.: Towards benchmarking scene background initialization. CoRR arxiv:1506.04051 (2015)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors certify that there is no conflict of interest with any individual/organization for the present work.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zhang, P., Geng, J., Liu, Y. et al. Robust principal component analysis based on tensor train rank and Schatten p-norm. Vis Comput 39, 5849–5867 (2023). https://doi.org/10.1007/s00371-022-02699-5
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00371-022-02699-5