^{1, 2}

^{1, 2}

^{1, 2}

^{1}

^{2}

Local Fisher discriminant analysis (LFDA) was proposed for dealing with the multimodal problem. It not only combines the idea of locality preserving projections (LPP) for preserving the local structure of the high-dimensional data but also combines the idea of Fisher discriminant analysis (FDA) for obtaining the discriminant power. However, LFDA also suffers from the undersampled problem as well as many dimensionality reduction methods. Meanwhile, the projection matrix is not sparse. In this paper, we propose double sparse local Fisher discriminant analysis (DSLFDA) for face recognition. The proposed method firstly constructs a sparse and data-adaptive graph with nonnegative constraint. Then, DSLFDA reformulates the objective function as a regression-type optimization problem. The undersampled problem is avoided naturally and the sparse solution can be obtained by adding the regression-type problem to a

Dimensionality reduction tries to transform the high-dimensional data into lower-dimensional space in order to preserve the useful information as much as possible. It has a wide range of applications in pattern recognition, machine learning, and computer vision. A well-known approach for supervised dimensionality reduction is linear discriminant analysis (LDA) [

To deal with the first problem, many methods have been proposed. Belhumeur et al. [

To deal with the second problem, many methods have been developed for dimensionality reduction. These methods focus on finding the local structure of the original data space. Locality preserving projections (LPP) [

To deal with the third problem, many dimensionality reduction methods integrating the sparse representation theory have been proposed. These methods can be classified into two categories. The first category focuses on finding a subspace spanned by sparse vectors. The sparse projection vectors reveal which element or region of the patterns is important for recognition tasks. Sparse PCA (SPCA) [

Motivated by

The rest of this paper is organized as follows. In Section

In this section, we give a brief of LDA and LFDA. Given a data set

The above transformation can be written as matrix form:

Linear discriminant analysis tries to find the discriminant vectors by the Fisher criterion, that is, the within-class distance is minimized and the between-class distance is maximized simultaneously. The within-class scatter matrix

Local Fisher discriminant analysis (LFDA) is also a discriminant analysis method. It aims to deal with the multimodal problem. The local within-class scatter matrix

The objection function of LFDA is formulated as

In LFDA, the affinity matrices are defined by the local scaling method. This method can be regarded as an extension of

We first reformulate formulas (

Matrices

The following result which was inspired by [

Suppose that

To obtain sparse projection vectors, we add a

Generally speaking, it is difficult to compute the optimal

For a fixed

The algorithm procedure of DSLFDA is summarized as follows.

Calculate affinity matrix

Calculate matrix

Initialize matrix

For given

Calculate the SVD of

Repeat steps 4 and 5 until converges.

In this section, we use the proposed DSLFDA method for face recognition. Three face image databases, that is, Yale [

The Yale face database contains 165 grayscale images of 15 individuals. Each individual has 11 images. These images were captured under lighting conditions (left-light, center-light, and right-light), with various facial expressions (normal, happy, sad, sleepy, surprised, and wink), and with facial details (with glasses or without). The original size of the images is

Sample images of two individuals from Yale database.

In the first experiment, we randomly select

The top recognition rates (%) and the corresponding dimensions on Yale database by different methods (mean ± std).

2 trains | 3 trains | 4 trains | 5 trains | 6 trains | |
---|---|---|---|---|---|

PCA | 46.30 ± 3.26 |
51.58 ± 4.00 |
56.19 ± 4.16 |
57.33 ± 4.60 |
62.40 ± 4.06 |

LDA | 45.11 ± 3.46 |
62.08 ± 4.31 |
70.86 ± 4.88 |
71.44 ± 5.19 |
77.22 ± 3.47 |

LPP | 45.26 ± 3.52 |
62.83 ± 4.22 |
70.57 ± 4.59 |
72.22 ± 3.81 |
78.00 ± 3.46 |

LFDA | 45.11 ± 3.46 |
62.50 ± 5.43 |
71.33 ± 5.07 |
72.33 ± 5.09 |
78.27 ± 3.72 |

SPCA | 43.19 ± 3.22 |
49.83 ± 4.08 |
54.95 ± 3.70 |
56.78 ± 3.33 |
61.87 ± 5.11 |

SLDA | 51.19 ± 5.78 |
63.85 ± 3.47 |
72.00 ± 4.76 |
72.56 ± 2.29 |
78.40 ± 2.42 |

SPP | 46.59 ± 5.36 |
52.92 ± 3.63 |
57.67 ± 3.54 |
58.48 ± 3.76 |
64.53 ± 4.63 |

DSNPE | 50.74 ± 5.26 |
63.58 ± 3.64 |
73.62 ± 5.04 |
75.89 ± 2.82 |
80.08 ± 2.61 |

DSLFDA | 53.11 ± 5.36 |
65.17 ± 3.68 |
73.24 ± 5.03 |
74.44 ± 3.51 |
81.47 ± 2.77 |

In the second experiment, we experiment with different dimensionalities of the projected space. Five images per individual were randomly selected for training, and the remaining images were used for testing. Figure

The recognition performance versus different dimensions on the Yale database.

The ORL database contains 400 images of 40 individuals. Each individual has 10 images. The images were captured at different times, under various light conditions, and with different facial expressions. The original size of the images is

Sample images of two individuals from ORL database.

In the first experiment, we randomly select

In the second experiment, we experiment with different dimensionalities of the projected space. Five images per individual were randomly selected for training, and the remaining images were randomly selected for testing. Figure

The top recognition rates (%) and the corresponding dimensions on ORL database by different methods (mean ± std).

2 trains | 3 trains | 4 trains | 5 trains | 6 trains | |
---|---|---|---|---|---|

PCA | 71.63 ± 3.06 |
79.14 ± 1.91 |
83.87 ± 1.77 |
86.85 ± 2.01 |
89.94 ± 2.72 |

LDA | 75.87 ± 3.42 |
85.96 ± 2.61 |
91.54 ± 1.51 |
93.85 ± 1.63 |
95.63 ± 1.89 |

LPP | 78.78 ± 2.80 |
86.25 ± 2.93 |
90.46 ± 1.83 |
94.40 ± 1.71 |
95.94 ± 1.96 |

LFDA | 82.13 ± 3.19 |
87.00 ± 2.97 |
90.83 ± 1.73 |
94.50 ± 1.90 |
96.06 ± 2.74 |

SPCA | 71.16 ± 2.80 |
78.50 ± 2.42 |
84.04 ± 1.82 |
86.80 ± 2.28 |
90.56 ± 2.03 |

SLDA | 82.03 ± 2.82 |
90.21 ± 1.85 |
93.79 ± 1.26 |
96.05 ± 1.83 |
97.56 ± 1.12 |

SPP | 79.53 ± 2.22 |
83.50 ± 2.08 |
86.58 ± 2.10 |
87.40 ± 2.63 |
88.19 ± 3.19 |

DSNPE | 78.97 ± 3.47 |
86.43 ± 1.92 |
92.92 ± 1.59 |
94.60 ± 1.96 |
95.94 ± 2.29 |

DSLFDA | 81.16 ± 2.88 |
90.25 ± 2.51 |
94.38 ± 1.54 |
96.65 ± 1.20 |
97.88 ± 1.03 |

The recognition performance versus different dimensions on the ORL database.

The CMU PIE face database contains 41368 images of 68 individuals. The images were captured under 13 different poses, under 43 different illumination conditions, and with 4 different expressions. In our experiments, we choose a subset (C29) that contains 1632 images of 68 individuals. These were manually cropped and resized to

Sample images of two individuals from CMU PIE database.

In the first experiment, we randomly select

The top recognition rates (%) and the corresponding dimensions on CMU PIE database by different methods (mean ± std).

3 trains | 6 trains | 9 trains | 12 trains | 15 trains | |
---|---|---|---|---|---|

PCA | 36.11 ± 0.85 |
55.05 ± 0.94 |
66.81 ± 2.08 |
76.80 ± 1.66 |
83.45 ± 1.77 |

LDA | 78.40 ± 1.43 |
87.70 ± 1.30 |
90.17 ± 1.04 |
91.67 ± 0.46 |
92.37 ± 0.56 |

LPP | 78.72 ± 1.19 |
89.40 ± 0.96 |
91.08 ± 0.86 |
92.22 ± 0.52 |
92.79 ± 0.65 |

LFDA | 78.68 ± 1.49 |
88.88 ± 1.09 |
90.70 ± 0.68 |
91.83 ± 0.51 |
92.43 ± 0.58 |

SPCA | 33.25 ± 0.60 |
52.51 ± 1.05 |
65.03 ± 2.14 |
75.40 ± 1.97 |
83.32 ± 2.02 |

SLDA | 78.17 ± 1.38 |
88.46 ± 1.23 |
92.15 ± 1.09 |
94.68 ± 0.36 |
96.36 ± 0.73 |

SPP | 75.11 ± 0.97 |
87.26 ± 1.04 |
90.59 ± 1.35 |
93.42 ± 0.56 |
95.36 ± 0.80 |

DSNPE | 78.93 ± 1.11 |
88.35 ± 1.22 |
91.77 ± 1.28 |
94.09 ± 0.60 |
96.05 ± 0.49 |

DSLFDA | 78.35 ± 1.24 |
88.99 ± 1.20 |
92.31 ± 1.10 |
95.10 ± 0.47 |
96.63 ± 0.62 |

In the second experiment, we experiment with different dimensionalities of the projected space. Fifteen images per individual were randomly selected for training, and the remaining images were randomly selected for testing. Figure

The recognition performance versus different dimensions on the CMU PIE database.

Based on the above experimental results, we can conclude the following observations.

For each method, the recognition rate increases with the increase of training sample sizes. The supervised extension of LPP can effectively improve the performance. PCA and SPCA achieve the worst results in all experiments; meanwhile, the performance of SPCA is inferior to that of PCA on all face databases. The reason may be that the number of nonzero variables for each component is selected equally.

For LDA and SLDA, the dimensionalities of projected subspace are

From Table

From the experimental results, we obtain that SPP can get competitive performance on CMU PIE database, rather than ORL and Yale databases. The reason may be that the sparse representation needs abundant training samples. Conversely, the nonnegative similarity measurement in DSLFDA is adaptive and can overcome the drawback of sparse representation.

DSNPE can be regarded as an extension of SPP. It can extract the discriminant information and perform better than SPP. On the Yale database, DSNPE can achieve the best recognition performance when the training samples per individual are four and five.

In this paper, we proposed a sparse projection method, called DSLFDA, for face recognition. It defines a novel affinity matrix that describes the relationships of points on the original high-dimensional data. The sparse projection vectors are obtained by solving the

We only focus on supervised learning in this paper. Because a large amount of unlabeled data is available in practical applications, semisupervised learning has attracted much attention in recent years [

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported partly by the National Natural Science Foundation of China (61172128 and 61003114), National Key Basic Research Program of China (2012CB316304), the Fundamental Research Funds for the Central Universities (2013JBM020 and 2013JBZ003), Program for Innovative Research Team in University of Ministry of Education of China (IRT201206), and Doctoral Foundation of China Ministry of Education (20120009120009).