A Multifactor Extension of Linear Discriminant Analysis for Face Recognition under Varying Pose and Illumination
© Sung Won Park and Marios Savvides. 2010
Received: 11 December 2009
Accepted: 20 May 2010
Published: 14 June 2010
Linear Discriminant Analysis (LDA) and Multilinear Principal Component Analysis (MPCA) are leading subspace methods for achieving dimension reduction based on supervised learning. Both LDA and MPCA use class labels of data samples to calculate subspaces onto which these samples are projected. Furthermore, both methods have been successfully applied to face recognition. Although LDA and MPCA share common goals and methodologies, in previous research they have been applied separately and independently. In this paper, we propose an extension of LDA to multiple factor frameworks. Our proposed method, Multifactor Discriminant Analysis, aims to obtain multilinear projections that maximize the between-class scatter while minimizing the withinclass scatter, which is the same core fundamental objective of LDA. Moreover, Multifactor Discriminant Analysis (MDA), like MPCA, uses multifactor analysis and calculates subject parameters that represent the characteristics of subjects and are invariant to other changes, such as viewpoints or lighting conditions. In this way, our proposed MDA combines the best virtues of both LDA and MPCA for face recognition.
Face recognition has significant applications for defense and national security. However, today, face recognition remains challenging because of large variations in facial image appearance due to multiple factors including facial feature variations among different subjects, viewpoints, lighting conditions, and facial expressions. Thus, there is great demand to develop robust face recognition methods that can recognize a subject's identity from a face image in the presence of such variations. Dimensionality reduction techniques are common approaches applied to face recognition not only to increase efficiency of matching and compact representation, but, more importantly, to highlight the important characteristics of each face image that provide discrimination. In particular, dimension reduction methods based on supervised learning have been proposed and commonly used in the following manner. Given a set of face images with class labels, dimension reduction methods based on supervised learning make full use of class labels of these images to learn each subject's identity. Then, a generalization of this dimension reduction is achieved for unlabeled test images, also called out-of-sample images. Finally, these test images are classified with respect to different subjects, and the classification accuracy is computed to evaluate the effectiveness of the discrimination.
Multilinear Principal Component Analysis (MPCA) [1, 2] and Linear Discriminant Analysis (LDA) [3, 4] are two of the most widely used dimension reduction methods for face recognition. Unlike traditional PCA, both MPCA and LDA are based on supervised learning that makes use of given class labels. Furthermore, both MPCA and LDA are subspace projection methods that calculate low-dimensional projections of data samples onto these trained subspaces. Although LDA and MPCA have different ways of calculating these subspaces, they have a common objective function which utilizes a subject's individual facial appearance variations.
MPCA is a multilinear extension of Principal Component Analysis (PCA)  that analyzes the interaction between multiple factors utilizing a tensor framework. The basic methodology of PCA is to calculate projections of data samples onto the linear subspace spanned by the principal directions with the largest variance. In other words, PCA finds the projections that best represent the data. While PCA calculates one type of low-dimensional projection vector for each face image, MPCA can obtain multiple types of low-dimensional projection vectors; each vector parameterizes a different factor of variations such as a subject's identity, viewpoint, and lighting feature spaces. MPCA establishes multiple dimensions based on multiple factors and then computes multiple linear subspaces representing multiple varying factors.
In this paper, we separately address the advantages and disadvantages of multifactor analysis and discriminant analysis and propose Multifactor Discriminant Analysis (MDA) by synthesizing both methods. MDA can be thought of as an extension of LDA to multiple factor frameworks providing both multifactor analysis and discriminant analysis. LDA and MPCA have different advantages and disadvantages, which result from the fact that each method assumes different characteristics for data distributions. LDA can analyze clusters distributed in a global data space based on the assumption that the samples of each class approximately create a Gaussian distribution. On the other hand, MPCA can analyze the locally repeated distributions which are caused by varying one factor under fixed other factors. Based on synthesizing both LDA and MPCA, our proposed MDA can capture both global and local distributions caused by a group of subjects.
Similar to our MDA, the Multilinear Discriminant Analysis proposed in  applies both tensor frameworks and LDA to face recognition. Our method aims to analyze multiple factors such as subjects' identities and lighting conditions in a set of vectored images. On the other hand,  is designed to analyze multidimensional images with a single factor, that is, subjects' identities. In , each face image constructs an -mode tensor, and the low-dimensional representation of this original tensor is calculated as another n-mode tensor with a smaller size. For example, if we simply use 2-mode tensors, that is, matrices, representing 2D images, the method proposed in  reduces each dimension of the rows and columns by capturing the repeated tendencies in rows and the repeated tendencies in columns. On the other hand, our proposed MDA analyzes the repeated tendencies caused by varying each factor in a subspace obtained by LDA. The goal of MDA is to reduce the impacts of environmental conditions, such as viewpoint and lighting, from the low-dimensional representations obtained by LDA. While  obtains a single tensor with a smaller size for each image tensor, our proposed MDA obtains multiple low-dimensional vectors, for each image vector, which decompose and parameterize the impacts of multiple factors. Thus, for each image, while the low-dimensional representation obtained by  is still influenced by variance in environmental factors, multiple parameters obtained by our MDA are expected to be independent from each other. The extension of  to multiple factor frameworks cannot be simply drawn because this method is formulated only using a single factor, that is to say, subjects' identities. On the other hand, our proposed MDA decomposes the low-dimensional representations obtained by LDA into multiple types of factor-specific parameters such as subject parameters.
The remainder of this paper is organized as follows. Section 2 reviews subspace methods from which the proposed method is derived. Section 3 first addresses the advantages and disadvantages of multifactor analysis and discriminant analysis individually, and then Section 4 proposes MDA with the combined virtues of both methods. Experimental results for face recognition in Section 5 show that the proposed MDA outperforms major dimension reduction methods on the CMU PIE database and the Extended Yale B database. Section 6 summarizes the results and conclusions of our proposed method.
2. Review of Subspace Projection Methods
In this section, we review MPCA and LDA, two methods on which our proposed Multifactor Discriminant Analysis is based.
2.1. Multilinear PCA
Multilinear Principal Component Analysis (MPCA) [1, 2] is a multilinear extension of PCA. MPCA computes a linear subspace representing the variance of data due to the variation of each factor as well as the linear subspace of the image space itself. In this paper, we consider three factors: different subjects, viewpoints (i.e., pose types), and lighting conditions (i.e., illumination). While PCA is based on Singular Value Decomposition (SVD) , MPCA is based on High-Order Singular Value Decomposition (HOSVD) , which is a multidimensional extension of SVD.
Since a Gram matrix is a matrix of all possible dot products, a set of also preserves the dot products of original training images.
These three Gram-like matrices , , , represent similarities between different subjects, different poses, and different lighting conditions, respectively. For example, can be thought of as the average similarity, measured by the dot product, between the th subject's face images and the th subject's face images under varying viewpoints and lighting conditions.
where denotes the Moore-Penrose pseudoinverse. To decompose the Kronecker product of multiple parameters into individual ones, two leading methods have been applied in  and . The best rank-1 method  reshapes the vector to the matrix , and using SVD of this matrix, is calculated as the left singular vector corresponding to the largest singular value. Another method is the rank- approximation using the alternating least squares method proposed in . In this paper, we employed the decomposition method proposed in , which produced slightly better performances for face recognition than the method proposed in .
Based on the observation that the Gram-like matrices in (8) are formulated using the dot products, Multifactor Kernel PCA (MKPCA), a kernel-based extension of MPCA, was introduced . If we define a kernel function , the kernel versions of the Gram-like matrices in (8) can be directly calculated. Thus, for training images, , , and can be also calculated using eigen decomposition of these matrices. Equations (10) and (11) show that in order to obtain , , and for any test image, also called an out-of-sample image, , we must be able to calculate and . Note that and are projections of training samples and a test sample onto nonlinear subspace, respectively, and these can be calculated by KPCA as shown in .
2.2. Linear Discriminant Analysis
where the columns of the matrix consist of . In other words, is the projection of onto the linear subspace spanned by . Note that . Despite the success of the LDA algorithm in many applications, the dimension of is often insufficient for representing each sample. This is caused by the fact that the number of available projection directions is lower than the class number . To improve this limitation of LDA, variants of LDA, such as the null subspace algorithm  and a direct LDA algorithm , were proposed.
3. Limitations of Multifactor Analysis and Discriminant Analysis
LDA and MPCA have different advantages and disadvantages, which result from the fact that each method assumes different characteristics for data distributions. MPCA's subject parameters represent the average positions of a group of subjects across varying viewpoints and lighting conditions. MPCA's averaging is premised on the assumption that these subjects maintain similar relative positions in a data space under each viewpoint and lighting condition. On the other hand, LDA is based on the assumption that the samples of each class approximately create a Gaussian distribution. Thus, we can expect that the comparative performances of MPCA and LDA vary with the characteristics of a data set. For classification tasks, LDA sometimes outperforms MPCA; at other times MPCA outperforms LDA. In this section, we demonstrate the assumptions on which each method is based and the conditions where one can outperform the other.
3.1. The Assumption of LDA: Clusters Caused by Different Classes
LDA inspires multiple advanced variants such as Kernel Discriminant Analysis (KDA) [14, 15], which can obtain nonlinear subspaces. However, these subspaces are still based on the analysis of the clusters distributed in a global data space. Thus, there is no guarantee that KDA can be successful if face images which belong to the same subject are scattered rather than distributed as clusters. In sum, LDA cannot be successfully applied unless, in a given data set, data samples are distributed as clusters due to different classes.
3.2. The Assumption of MPCA: Repeated Distributions Caused by Varying One Factor
In Euclidean geometry, the dot product between two vectors formulates the distance and linear similarity between them. Equation (9) shows that is also the Gram matrix of a set of the column vectors of the matrix . Thus, these column vectors represent the average distances between pairs of subjects. Therefore, the row vectors of , that is, the subject parameters, depend on these average distances between subject across varying viewpoints and lighting conditions. Similarly, the viewpoint parameters and the lighting parameters depend on the average distances between viewpoints and lighting conditions, respectively, in a data space.
Based on the above expectations, if varying just one factor generates dissimilar shapes of distribution, multilinear subspaces based on these average shapes do not represent a variety of data distributions. In Figure 3(a), some curves have W-shapes while most of the other curves have V-shapes. Thus, in this case, we cannot expect reliable performances from MPCA because the average shape obtained by MPCA for each factor insufficiently covers individual shapes of curves.
4. Multifactor Discriminant Analysis
As shown in Section 3.1, for face recognition, LDA is preferred if in a given data set, face images are distributed as clusters due to different subjects. Unlike LDA, as shown in Section 3.2, MPCA can be successfully applied to face recognition if various subjects' face images repeat similar shapes of distributions under each viewpoint and lighting, even if these subjects do not seem to create these clusters. In this paper, we propose a novel method which can offer the advantages of both methods. Our proposed method is based on an extension of LDA to multiple factor frameworks. Thus, we can call our method Multifactor Discriminant Analysis (MDA). From , MDA aims to remove the remaining characteristics which are caused by other factors, such as viewpoints and lighting conditions.
where is the low-dimensional representation of obtained by PCA. Thus, we can think that performs a linear transformation which maps the Kronecker product of multiple factor-specific parameters to the low-dimensional representations provided by PCA. In other words, is decomposed into , , and by using the transformation matrix .
where is the LDA transformation matrix defined in (14) and (15). As reviewed in Section 2.2, , the number of available projection directions, is lower than the class number : . Note that in (20) is formulated in a similar way to in (19) using different factor-specific parameters and . We expect in (20), the subject parameter obtained by MDA, to be more reliable than both and since provides the advantages of the virtues of both LDA and MPCA. Using (15), we also calculate the matrix whose columns are the LDA projections of training samples.
To obtain the factor-specific parameters of an arbitrary test image , we perform the following steps. During training, we first calculate the three orthogonal matrices, , , and , and subsequently . Then, during testing, for the LDA projection of an arbitrary test image, we calculate the factor-specific parameters by decomposing .
where denotes the LDA projection of a training image of the th subject under the th viewpoint and the th lighting condition. In (9), for MPCA, , , and are calculated as the eigenvector matrices of , , and , respectively. In similar ways, for MDA, , , and can be calculated as the eigenvector matrices of , , and , respectively. Again, each row vector of represents the subject parameter of each subject in a training set.
this matrix does not have full column rank. If is decomposed by SVD, has nonzero singular values at most. However, each of the matrices , , and has full column rank since these matrices are defined in terms of the averages of different parts of as shown in Figure 4. Thus, even if or , one can calculate valid , , and eigenvectors from , , and , respectively.
Again, as done in (11), by SVD of the matrix , is calculated as the left singular vector corresponding to the largest singular value. Consequently, we can obtain of an arbitrary image test .
5. Experimental Results
In this section, we demonstrate that Multifactor Discriminant Analysis is an appropriate method for dimension reduction of face images with varying factors. To test the quality of dimension reduction, we conducted face recognition tests. In all experiments, face images are aligned using eye coordinates and then cropped. Then, face images were resized to gray-scale images, and each vectored image was normalized with unit norm and zero mean. After aligning and cropping, the left and right eyes are located at and , respectively, in each image.
Rank-1 recognition rate on the Extended YaleB database.
viewpoints & lighting
Rank 1 recognition rate on the CMU PIE database.
viewpoints & lighting
We compare the performance of our proposed method, Multifactor Discriminant Analysis, and other traditional subspace projection methods with respect to dimension reduction: PCA, MPCA, KPCA, and LDA. For PCA and KPCA, we used the subspaces consisting of the minimum numbers of eigenvectors whose cumulative energy is above 0.95. For MPCA, we set the threshold in pixel mode to 0.95 and the threshold in other modes to 1.0. KPCA used RBF kernels with set to . We compared the rank-1 recognition rates of all of the methods using the simple cosine distance.
As shown in Tables 1 and 2, our proposed method, Multifactor Discriminant Analysis, outperforms the other methods for face recognition. This seems to be because Multifactor Discriminant Analysis offers the combined virtues of both multifactor analysis methods and discriminant analysis methods. Like multilinear subspace methods, Multifactor Discriminant Analysis can analyze one sample in a multiple factor framework, which improves face recognition performance.
In this paper, we propose a novel dimension reduction method for face recognition: Multifactor Discriminant Analysis. Multifactor Discriminant Analysis can be thought of as an extension of LDA to multiple factor frameworks providing both multifactor analysis and discriminant analysis. Moreover, we have shown through experiments that MDA extracts more reliable subject parameters compared to the low-dimensional projections obtained by LDA and MPCA. These subject parameters obtained by MDA represent locally repeated shapes of distributions due to differences in subjects for each combination of other factors. Consequently, MDA can offer more discriminant power, making full use of both global distribution of the entire data set and local factor-specific distribution. Reference  introduced another method which is theoretically based on both MPCA and LDA: Multilinear Discriminant Analysis. However, Multilinear Discriminant Analysis cannot analyze multiple factor frameworks, while our proposed Multifactor Discriminant Analysis can. Relevant examples are shown in Figure 5 where our proposed approach has been able to yield a discriminative two dimensional subspace that can cluster multiple subjects in the Yale-B database. On the other hand, LDA completely spreads the data samples into one global undiscriminative distribution of data samples. These results show the dimension reduction power of our approach in the presence of nuisance factors such as viewpoints and lighting conditions. This improved dimension reduction power will allow us to have reduced size feature sets (optimal for template storage) and increased matching speed due to these smaller dimensional features. Our approach is thus attractive for robust face recognition for real-world defense and security applications. Future work will include evaluating this approach on larger data sets such as the CMU Multi-PIE database and NIST's FRGC and MBGC databases.
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