The Visual Computer

, Volume 32, Issue 9, pp 1165–1178 | Cite as

Fisher discrimination-based \(l_{2,1} \)-norm sparse representation for face recognition

  • Lu ZhaoEmail author
  • Yong Zhang
  • Baocai Yin
  • Yanfeng Sun
  • Yongli Hu
  • Xinglin Piao
  • Qianjun Wu
Original Article


In recent years, sparse representation-based classification (SRC) has made great progress in face recognition (FR). However, SRC emphasizes noise sparsity too much and it is not suitable for the real world. In this paper, we propose a robust \(l_{2,1}\)-norm Sparse Representation framework that constrains the noise penalty by the \(l_{2,1}\)-norm. The \(l_{2,1} \)-norm takes advantage of both the discriminative nature of the \(l_1 \)-norm and the systemic representation of the \(l_2 \)-norm. In addition, we use the nuclear norm to constrain the coefficient matrix. Motivated by the Fisher criterion, we propose the Fisher discriminant-based \(l_{2,1} \)-norm sparse representation method for FR which utilizes a supervised approach. Thus, we consider the within-class scatter and between-class scatter when all of the label information is available. The paper shows that the model can provide stronger discriminant power than the classical sparse representation models and can be solved by the alternating direction method of multiplier. Additionally, it is robust to the contiguous occlusion noise. Extensive experiments demonstrate that our method achieves significantly better results than SRC and some other sparse representation methods for FR when addressing large regions with contiguous occlusion.


Sparse representation \(l_{2 , 1}\)-Norm Face recognition Fisher discriminant 



The research project was supported by the National Natural Foundation of China under Grant No. 61390510, 61300065, 61370119, 61171169 and Beijing Natural Science Foundation No. 4132013, 4142010 and supported by the Beijing science and technology project No. Z151100002115040, and also supported by PHR(IHLB).


  1. 1.
    Zhao, W., Chellappa, R., Phillips, P., Rosenfeld, A.: Face recognition: a literature survey. ACM Comput. Surv. 35(4), 399–458 (2003)CrossRefGoogle Scholar
  2. 2.
    Bartlett, M., Movellan, J., Sejnowski, T.: Face recognition by independent component analysis. IEEE Trans. Neural Netw. 13(6), 1450–1464 (2002)CrossRefGoogle Scholar
  3. 3.
    Belhumeur, P., Hespanha, J., Kriegman, D.: Eigenfaces vs. fisherfaces: recognition using class specific linear projection. IEEE Trans. Pattern Anal. Mach. Intell. 19(7), 711–720 (1997)CrossRefGoogle Scholar
  4. 4.
    Turk, M., Pentland, A.: Eigenfaces for recognition. J. Cognit. Neurosci. 3(1), 71–86 (1991)CrossRefGoogle Scholar
  5. 5.
    Zhang, B., Gao, Y., Zhao, S., Liu, J.: Local derivative pattern versus local binary pattern: face recognition with high-order local pattern descriptor. IEEE Trans. Image Process. 19(2), 533–544 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Tan, X., Triggs, B.: Enhanced local texture feature sets for face recognition under difficult lighting conditions. IEEE Trans. Image Process. 19(6), 1635–1650 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ahonen, T., Hadid, A., Pietikainen, M.: Face description with local binary patterns: application to face recognition. IEEE Trans. Pattern Anal. Mach. Intell. 28(12), 2037–2041 (2006)CrossRefzbMATHGoogle Scholar
  8. 8.
    Ortiz, E.G., Wright, A., Shah, M.: Face recognition in movie trailers via mean sequence sparse representation-based classification. In: Proceedings of the IEEE Conference CVPR, pp. 3531–3538 (2013)Google Scholar
  9. 9.
    Ocegueda, O., Fang, T., Shah, S., Kakadiaris, I.: 3D-face discriminant analysis using Gauss–Markov posterior marginals. IEEE Trans. Pattern Anal. Mach. Intell. 35(3), 728–739 (2013)CrossRefGoogle Scholar
  10. 10.
    Quan, W., Jiang, Y., Zhang, J., Chen, J.X.: Robust object tracking with active context learning. Vis. Comput. 31(10), 1307–1318 (2015)CrossRefGoogle Scholar
  11. 11.
    Dawn, D.D., Shaikh, S.H.: A comprehensive survey of human action recognition with spatio-temporal interest point (STIP) detector. Vis. Comput. 1–18 (2015). doi: 10.1007/s00371-015-1066-2
  12. 12.
    Wu, J.Z., Hu, D., Chen, F.L.: Action recognition by hidden temporal models. Vis. Comput. 30(12), 1395–1404 (2013)CrossRefGoogle Scholar
  13. 13.
    Hong, J., Aleix, M.: Support vector machines in face recognition with occlusions. In: Conference on Computer Vision and Pattern Recognition, IEEE, pp. 136–141 (2009)Google Scholar
  14. 14.
    Tan, X., Chen, S., Zhou, Z.H., Liu, J.: Face recognition under occlusions and variant expressions with partial similarity. IEEE Trans. Inf. Forens. Secur. 4(2), 217–230 (2009)CrossRefGoogle Scholar
  15. 15.
    Lee, D., Seung, H., et al.: Learning the parts of objects by non-negative matrix factorization. Nature 401(6755), 788–791 (1999)CrossRefGoogle Scholar
  16. 16.
    Yuen, P., Lai, J.: Face representation using independent component analysis. Pattern Recognit. 35(6), 1247–1257 (2002)CrossRefzbMATHGoogle Scholar
  17. 17.
    Liu, C., Wechsler, H.: Gabor feature based classification using the enhanced fisher linear discriminant model for face recognition. IEEE Trans. Image Process. 11(4), 467–476 (2002)CrossRefGoogle Scholar
  18. 18.
    Zhou, Z., Wagner, A., Mobahi, H., Wright, J., Ma, Y.: Face recognition with contiguous occlusion using Markov random fields. In: International Conference on Computer Vision, IEEE, pp. 1050–1057 (2009)Google Scholar
  19. 19.
    Olshausen, B.A., Field, D.J.: Sparse coding with an over-complete basis set: a strategy employed by v1? Vis. Res. 37(23), 3311–3325 (1997)CrossRefGoogle Scholar
  20. 20.
    Vinje, W.E., Gallant, J.L.: Sparse coding and decorrelation in primary visual cortex during natural vision. Science 287(5456), 1273–1276 (2000)CrossRefGoogle Scholar
  21. 21.
    Yuan, X., Yan, S.: Visual classification with multi-task joint sparse representation. In: 2010 IEEE conference on computer vision and pattern recognition (CVPR), vol. 21, pp. 3493–3500. IEEE, San Francisco, CA (2010)Google Scholar
  22. 22.
    Jenatton, R., Mairal, J., Obozinski, G., Bach, F.: Proximal methods for hierarchical sparse coding. J. Mach. Learn. Res. 12, 2297–2334 (2011)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Chao, Y.W., Yeh, Y.R., Chen, Y.W., Lee, Y.J., Wang, Y.C.F.: Locality-constrained group sparse representation for robust face recognition. In: 18th IEEE international conference on image processing (ICIP), vol. 263, pp. 761–764. IEEE, Brussels (2011)Google Scholar
  24. 24.
    Wright, J., Yang, A., Ganesh, A., Sastry, S., Ma, Y.: Robust face recognition via sparse representation. IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 210–227 (2009)CrossRefGoogle Scholar
  25. 25.
    Wright, J., Ma, Y., Mairal, J., Ssairo, G., Huang, T., Yan, S.C.: Sparse representation for computer vision and pattern recognition. Proc. IEEE 98(6), 1031–1044 (2010)CrossRefGoogle Scholar
  26. 26.
    Yang, M., Zhang, L., Yang, J., Zhang, D.: Robust sparse coding for face recognition. In: 2011 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), vol. 47, pp. 625–632. IEEE, Providence, RI (2011)Google Scholar
  27. 27.
    Huang, K., Aviyente, S.: Sparse representation for signal classification. Adv. Neural Inf. Process. Syst. 19, 609–616 (2007)Google Scholar
  28. 28.
    Wagner, A., Wright, J., Ganesh, A., Zhou, Z.H., Ma, Y.: Towards a practical face recognition system: robust registration and illumination by sparse representation. In: 2009 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), vol. 34, pp. 597–604. IEEE, Miami, FL (2009)Google Scholar
  29. 29.
    Candes, E.J., Li, X.D., Ma, Y., Wright, J.: Robust principal component analysis? J. ACM 58(1), 1–37 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wanger, A., Wright, J., Ganesh, A., Zhou, Z.H., Ma, Y.: Towards a practical face recognition system: robust registration and illumination by sparse representation. IEEE Pattern Anal. Mach. Intell. 34(2), 372–386 (2012)CrossRefGoogle Scholar
  31. 31.
    Wang, J., Lu, C.Y., Wang, M., Li, P.P.: Robust face recognition via adaptive sparse representation. IEEE Trans. Cybern. 44(12), 2368–2378 (2014)Google Scholar
  32. 32.
    Edouard, G., Guillaume, O., Francis, B.: Trace lasso: a trace norm regularization for correlated designs. In: Proceedings of the Advances in Neural Information Processing Systems, pp. 2187–2195 (2011)Google Scholar
  33. 33.
    Ou, W.H., You, X.G., Tao, D.C., Zhang, P.Y., et al.: Robust face recognition via occlusion dictionary learning. Pattern Recognit. 47, 1559–1572 (2014)CrossRefGoogle Scholar
  34. 34.
    Lee, H., Battle, A., Raina, R., Ng, A.Y.: Efficient sparse coding algorithms. Adv. Neural Inf. Process. Syst. 19, 801–808 (2007)Google Scholar
  35. 35.
    Zhang, Z.L., Yu, M., Jia, J., Liu, H., et al.: Fisher discriminant based low rank matrix recovery for face recognition. Pattern Recognit. 47, 3502–3511 (2014)CrossRefGoogle Scholar
  36. 36.
    Liu, G., Lin, Z., Yu, Y.: Robust subspace segmentation by low-rank representation. In: Proceedings of the ICML, pp. 663–670 (2010)Google Scholar
  37. 37.
    Belhumeur, P.N., Hespanha, J.P., Kriegman, D.J.: Eigenfaces vs fisherface: recognition using class specific linear projection. IEEE Pattern Anal. Mach. Intell. 19(7), 711–720 (1997)CrossRefGoogle Scholar
  38. 38.
    Li, Z., Lin, D., Tang, X.: Nonparametric discriminant analysis for face recognition. IEEE Pattern Anal. Mach. Intell. 31(4), 755–761 (2009)CrossRefGoogle Scholar
  39. 39.
    Lu, J., Tan, Y., Wang, G.: Discriminative multi-manifold analysis for face recognition from a single training sample per person. IEEE Pattern Anal. Mach. Intell. 35(1), 39–51 (2013)Google Scholar
  40. 40.
    Hua, G., Viola, P., Drucker, S.: Face recognition using discriminatively trained orthogonal rank one tensor projections. In: IEEE Conference on Computer Vision and Pattern Recognition, 2007, CVPR ’07, vol. 7, pp. 1–8. IEEE, Minneapolis, MN (2007)Google Scholar
  41. 41.
    Gao, Q., Liu, J., Zhang, H., Hou, J., Yang, X.: Enhanced fisher discriminant criterion for image classification. Pattern Recognit. 45(10), 3717–3724 (2012)CrossRefGoogle Scholar
  42. 42.
    Gao, Q., Ma, J., Zhang, H., Gao, X., Liu, Y.: Stable orthogonal discriminant embedding for linear dimensionality reduction. IEEE Trans. Image Process. 22(7), 2521–2531 (2013)CrossRefGoogle Scholar
  43. 43.
    Yang, J., Chu, D., Zhang, L.: Sparse representation classifier steered discriminative projection with applications to face recognition. IEEE Trans. Neural Netw. Learn. Syst. 24(7), 1023–1035 (2013)CrossRefGoogle Scholar
  44. 44.
    Zhang, N., Yang, J.: Low-rank representation based discriminative projection for robust feature extraction. Neurocomputing 111(2), 13–20 (2013)CrossRefGoogle Scholar
  45. 45.
    Ding, C., Zhou, D., He, X.F., Zha, H.Y.: R1-PCA: rotational invariant L1-norm principal component analysis for robust subspace factorization. In: Proceeding of the 23th International Conference on Machine Learning, pp. 281–288 (2006)Google Scholar
  46. 46.
    Liu, G., Lin, Z., Yan, S., Sun, J., Yu, Y., Ma, Y.: Robust recovery of subspace structures by low-rank representation. IEEE Pattern Anal. Mach. Intell. 35(1), 171–184 (2013)CrossRefGoogle Scholar
  47. 47.
    Belhumeur, P.N., Hespanha, J.P., Kriegman, D.J.: Eigenfaces vs fisherfaces: recognition using class specific linear projection. IEEE Pattern Anal. Mach. Intell. 19(7), 711–720 (1997)CrossRefGoogle Scholar
  48. 48.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  49. 49.
    Qian, J.J., Yang, J., Zhang, F.L., Lin, Z.C.: Robust low-rank regularized regression for face recognition with occlusion. In: Computer Vision and Pattern Recognition Workshops, pp. 21–26 (2014)Google Scholar
  50. 50.
    Yang, J., Zhang, W.Y., Wang, Y.: A fast algorithm for edge-preserving variational multichannel image restoration. SIAM J. Imaging Sci. 2(2), 569–592 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Cai, J.F., Candes, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956–1982 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Yang, M., Zhang, L., Feng, X.C., Zhang, D.: Fisher discrimination dictionary learning for sparse representation. In: 2011 IEEE International Conference on Computer Vision (ICCV), pp. 543–550. IEEE, Barcelona (2011)Google Scholar
  53. 53.
    Can, C.Y., Min, H., Gui, J., Zhu, L., Lei, Y.K.: Face recognition via weighted sparse representation. J. Vis. Commun. Image R 24, 111–116 (2013)CrossRefGoogle Scholar
  54. 54.
    Lee, K., Ho, J., Kriegman, D.: Acquiring linear subspaces for face recognition under variable lighting. IEEE Trans. Pattern Anal. Mach. Intell. 27(5), 684–698 (2005)CrossRefGoogle Scholar
  55. 55.
    Georghiades, A., Belhumeur, P., Kriegman, D.: From few to many: illumination cone models for face recognition under variable lighting and pose. IEEE Trans. Pattern Anal. Mach. Intell. 23(6), 643–660 (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Lu Zhao
    • 1
    Email author
  • Yong Zhang
    • 1
  • Baocai Yin
    • 1
    • 2
    • 3
  • Yanfeng Sun
    • 1
  • Yongli Hu
    • 1
  • Xinglin Piao
    • 1
  • Qianjun Wu
    • 1
  1. 1.Beijing Key Laboratory of Multimedia and Intelligent Software Technology, College of Metropolitan TransportationBeijing University of TechnologyBeijingChina
  2. 2.School of Software TechnologyDalian University of TechnologyDalianChina
  3. 3.Collaborative Innovation Center of Electric Vehicles in BeijingBeijingChina

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