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

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Abstract

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.

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Acknowledgments

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).

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Authors and Affiliations

  1. Beijing Key Laboratory of Multimedia and Intelligent Software Technology, College of Metropolitan Transportation, Beijing University of Technology, Beijing, 100124, China

    Lu Zhao, Yong Zhang, Baocai Yin, Yanfeng Sun, Yongli Hu, Xinglin Piao & Qianjun Wu

  2. School of Software Technology, Dalian University of Technology, Dalian, 116024, China

    Baocai Yin

  3. Collaborative Innovation Center of Electric Vehicles in Beijing, Beijing, China

    Baocai Yin

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  1. Lu Zhao
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Correspondence to Lu Zhao.

Appendix

Appendix

Definitions of \(R_1,R_2,Q,{\hat{Q}}\).

Let \(E_i =[1]_{n_i \times n_i } \) be a matrix of size \(n_i \times n_i \) with all entries being 1, and let \(R_1 =I_{n\times n} -\hbox {diag}(E_i)/n\), \(i=[1,2,\ldots ,c]\); \(R_2 =1/n\,\hbox {diag}(E_i )-1/{nc}\,I_{n\times n};\) \(R(A)=\sum \nolimits _{i=1}^c {\sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^c {\Vert {D_j A_i^j}\Vert }_F^2 }, A_i^j\) is the coding coefficient of \(Y_i\) over the sub-dictionary \(D_j\), \(A_i^j\) should have nearly zero coefficients, and thus, we use a select matrix Q with a dot product for the constraint. To facilitate the calculation, we use each row vector of the matrix Q as a diagonal, expanding to a diagonal matrix \({\hat{Q}}\).

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Zhao, L., Zhang, Y., Yin, B. et al. Fisher discrimination-based \(l_{2,1} \)-norm sparse representation for face recognition. Vis Comput 32, 1165–1178 (2016). https://doi.org/10.1007/s00371-015-1169-9

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