In pattern recognition, feature extraction techniques have been widely employed to reduce the dimensionality of highdimensional data. In this paper, we propose a novel feature extraction algorithm called membershipdegree preserving discriminant analysis (MPDA) based on the fisher criterion and fuzzy set theory for face recognition. In the proposed algorithm, the membership degree of each sample to particular classes is firstly calculated by the fuzzy
In the fields of computer vision and pattern recognition, many applications such as face recognition often suffer from the highdimensional problem. In recent years, feature extraction techniques have been widely employed to reduce the dimensionality of highdimensional data. So far, there have been a variety of feature extraction techniques. Among them, Principal Component Analysis (PCA) [
Recently, manifold learning based algorithms which are straightforward in detecting the nonlinear structures have attracted much attention of the researchers. The representative algorithms include Locally Linear Embedding (LLE) [
Motivated by LDA and some other manifold learning based algorithms, this paper presents a new method called MembershipDegree Preserving Projection Analysis (MPDA) for feature extraction. MPDA utilizes the fuzzy
The remainder of this paper is organized as follows. Section
Given a set of
LDA seeks to find a set of projection directions such that the fisher criterion (i.e., the ratio of the betweenclass scatter to the withinclass scatter) is maximized after projection of samples. Thus the objective function of LDA can be defined by the following equation:
In order to find out
From the literature [
According to (
Sample membership indicates its dependence on a certain class. Defining
Compute the Euclidean distance matrix between pairs of feature vectors in training set.
Set diagonal elements of this Euclidean distance matrix to infinity.
Sort the distance matrix (treat each of its columns separately) in an ascending order. Collect the corresponding class labels of the patterns located in the closest neighborhood of the pattern under consideration (as we are concerned with “
Compute the membership degree to class “
In the above expression,
Therefore, the fuzzy membership matrix
In MPDA, the withinclass scatter matrix and the betweenclass scatter matrix of samples can separately be described as
Similar to LDA, MPDA also attempts to find an optimal projection vector such that the betweenclass scatter is maximized and the withinclass scatter is minimized after projection of samples. That is to say, the objective function of MPDA should have the following form:
In order to seek out
The algorithm of MPDA can be described as follows.
For the
According to (
Computing the corresponding feature vectors
Sample
After obtaining the representation
To demonstrate the effectiveness of the proposed algorithm, MPDA, PCA, LDA, LPP, UDP, and MFA are evaluated on the ORL, Yale, and FERET face databases. After implementing the algorithms for feature extraction, the nearest neighbor classifier with Euclidean distance as a distance measure is used for classification.
As human face image can be up to millions of dimensions, while human’s vision is up to 3 dimensions at best, in order to more visually figure out the internal connection between the data, to compare the difference and performance of these algorithms, in the research, the four algorithms of MPDA, LPP, UDP, and MFA are selected to take image twodimensional visualization test. The ORL database (
10 images of one person on the ORL database.
In this experiment, we select the images of the first five persons in the ORL database for visualization, thus the total images are 10 × 5 = 50. Let “*,” “○,” “+,” “△,” and “
Twodimensional projection results of LPP, UDP, MFA, and MPDA.
LPP
UDP
MFA
MPDA
In order to evaluate and verify the recognition performance of MPDA compared with other methods such as PCA, LDA, LPP, UDP, and MFA, the proposed algorithm is implemented on the Yale and FERET face databases. The Yale face database (
11 images of one person on the Yale database.
The FERET database (
Seven images of one person in the FERET database.
On the Yale database, the first
The maximal recognition rates (%) and corresponding dimensions (shown in the parentheses) of MPDA with different
Training number 






4  91.43 (14)  95.24 (14)  97.14 (16)  96.19 (12)  96.19 (12) 
5  92.22 (16)  96.67 (16)  98.67 (15)  97.33 (14)  97.33 (15) 
The maximal recognition rates (%) and corresponding dimensions (shown in the parentheses) of MPDA compared with other algorithms on the Yale database.
Training number  PCA  LDA  LPP  UDP  MFA  MPDA 

4  91.43 (21)  96.19 (18)  95.24 (17)  96.19 (19)  96.19 (15) 

5  92.22 (25)  95.56 (18)  96.67 (23)  96.67 (24)  97.78 (29) 

On the FERET database, the first
The maximal recognition rates (%) and corresponding dimensions (shown in the parentheses) of MPDA compared with other algorithms on the FERET database.
Training number  PCA  LDA  LPP  MFA  UDP  MPDA 

3  60.25 (60)  72.75 (10)  61.50 (40)  72.80 (50)  69.75 (20) 

4  70.25 (60)  77.75 (10)  71.50 (40)  80.88 (50)  79.75 (20) 

Recognition rates and corresponding dimensions of MPDA and other algorithms on the FERET database when
Recognition rates and corresponding dimensions of MPDA and other algorithms on the FERET database when
This paper developed a novel method MPDA for face recognition. MPDA significantly describe the internal manifold structure of the sample. In MPDA, we use the fuzzy
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Natural Science Fund of China (Grant nos. 60632050, 9082004, 61202318, 61373062, and 61373063), the National 863 Project (Grant no. 2006AA04Z238), and the Basic Key Technology Project of the Ministry of Industry and Information Technology of China (Grant no. E0310/1112/JC01).