A new framework to combine vertical and horizontal information for face recognition
Introduction
In the area of face recognition (FR), vector-based models are the most popular. Under this model paradigm, the original two-dimensional (2D) image is transformed into a one-dimensional (1D) vector, and can be presented as a point in the high-dimensional image space [2], [13], [14]. Many 1D-based algorithms such as Principal Component Analysis (PCA) [13], [14] and Linear Discriminant Analysis (LDA) [1], [29] adopt this model and numerous successes have been made. As a well-known feature extraction and data representation technique, PCA is widely used in the areas of pattern recognition. It seeks a subspace that best represents the data in a least-squares sense, so the features extracted by PCA are called the most expressive features [12]. PCA-based algorithms are popular because of the ease of implementation and reasonable performance [10], [11]. LDA is a supervised technique for dealing with classification problems. The subspace with the most discriminative information can be gained through LDA by maximizing the ratio of the between-class scatter to within-class scatter. The features extracted by LDA are called the most discriminating features [12], and LDA is then proved to be more efficient than PCA [1], [29]. Compared with unsupervised methods PCA, LDA is prone to encounter “small sample size” (3S) problem, which widely exists in the FR tasks where the number of training samples is smaller than the dimensionality of the samples [1], [3], [27]. In addition, reshaping 2D image into 1D vector leads to very high-dimensional image vector, where the scatter matrix is difficult to be evaluated due to its large size, which is well known as curse of the dimensionality dilemma.
Recently, lots of works in 2D and tensor subspace analysis [4], [8], [9], [16], [17], [18], [19], [20], [21], [24], [25], [26], which aim at addressing these problems have been reported, among which 2DLDA [4], [25], [26] is the typical one. 2DLDA directly computes eigenvectors of the so-called scatter matrices without reshaping the matrix into vector. Thus, the difficulty resulting from the large size of scatter matrices is efficiently and artfully avoided. It was reported in Ref. [8] that the recognition accuracy on several databases was higher by using 2DLDA than by other PCA- and LDA-based algorithms. Nonetheless, a main drawback of 2DLDA is that it needs more coefficients than LDA for image presentation, and thus requires more memory to store the features. The idea of 2DLDA was developed and two-directional 2DLDA ((2D)2LDA) [9], [24] was proposed to overcome the disadvantage of 2DLDA. The main idea of (2D)2LDA is to perform 2DLDA twice: the first in the horizontal direction and then in the vertical direction. After (2D)2LDA, the discriminant information is compacted into the up-left corner of the image.
The work of Wang et al. [15] indicates that performing 2DLDA in the horizontal direction is equivalent to performing LDA on the rows of the image if each line (row) is viewed as a computational unit. We believe that 2DLDA in the horizontal direction considers the discriminative information on the rows, but neglects partial vertical information. Thus the following step of 2DLDA in the vertical direction cannot utilize the entire vertical discriminative information available to the image. That is to say, the proposed (2D)2LDA does not use both horizontal and vertical information equally but emphasizes extracting horizontal discriminant features. In this article we present a two-dimensional combined complex discriminant analysis (2DCCDA) framework. In the proposed method, 2DLDA in the horizontal and vertical direction are firstly performed separately, and then these two real feature matrices are integrated into a complex feature matrix. A complex version of 2DLDA is introduced to extract the discriminant complex features of this complex feature matrix for feature selection.
The paper is organized as follows. 2DLDA is reviewed and our new idea is introduced in Section 2. In Section 3, experiments are presented to demonstrate the efficiency of our proposed methods. Finally the conclusions are given in Section 4.
Section snippets
Outline of 2DLDA
Assume there is training set (p=1,2,…,C, q=1,2,…,Np), where C is the number of class, and the pth class has Np training samples. In class p, the qth training image is denoted by an m×n image matrix . The mean image of class p and mean image of all training samples are denoted by and , respectively.
The image between-class scatter matrix Sb and image within-class scatter matrix Sw can be constructed bywhere T denotes
Experiment
We have investigated the performance of our proposed 2DCCDA method for FR on two benchmark databases: AT&T and AR databases. While the AR database is used to examine the performance of the algorithms under the condition of varied facial expression and lighting configuration, the AT&T database is used to test the performance of the FR algorithms under conditions where the pose and sample size are varied. The results of the following standard methods are also provided for comparison: (1)
Conclusions
The (2D)2LDA uses 2DLDA to extract horizontal and vertical features sequentially. Thus it leans to extracting horizontal discriminative features. The alternative (2D)2LDA, a transformative form of (2D)2LDA, emphasizes extracting vertical information. We integrated their excellences and developed an efficient feature extraction framework named two-dimensional combined complex discriminant analysis (2DCCDA) to remedy the drawback. In this framework, 2DLDA is first performed vertically and then
Acknowledgments
The authors would like to express heartfelt gratitude to 973 Project (2005CB724303) of China for the financial support and our sincere thanks to our colleagues for their continuous and generous support.
Wangxin Yu received bachelor degree in Biomedical Engineering from Shanghai Jiaotong University, PR China, in 2003, where he is currently working toward the Ph.D. degree. His research interests include bio-signal processing, face detection and pattern recognition.
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Wangxin Yu received bachelor degree in Biomedical Engineering from Shanghai Jiaotong University, PR China, in 2003, where he is currently working toward the Ph.D. degree. His research interests include bio-signal processing, face detection and pattern recognition.
Zhizhong Wang is a Professor of Biomedical Engineering at Shanghai Jiaotong University, China, and serving as a Member of National Medical Electrical technology standardization committee, director of Medical instrument of the branch of China Instrumentation Society and Committee member of Shanghai Academic Degree Committee. His research fields include medical signal processing and its applications combining with computer technology, modeling of human body and medical signal analyzing by modern control theory and signal processing methodology.
Weiting Chen received the M.S. degree in Biomedical Engineering in 2004 from Shanghai Jiaotong University, Shanghai, China, where she is currently working toward the Ph.D. degree. Her research interests include nonlinear time series analysis, biomedical signal processing and pattern recognition.