重庆邮电大学本科毕业设计（论文）
附 录
一、英文原文
An improved face recognition technique based
on modular PCA approach
Rajkiran Gottumukkal Vijayan K.Asari
Abstract
A face recognition algorithm based on modular PCA approach is presented in this paper. The proposed algorithm when compared with conventional PCA algorithm has an improved recognition rate for face images with large variations in lighting direction and facial expression. In the proposed technique, the face images are divided into smaller sub-images and the PCA approach is applied to each of these sub-images. Since some of the local facial features of an individual do not vary even when the pose, lighting direction and facial expression vary, we expect the proposed method to be able to cope with these variations. The accuracy of the conventional PCA method and modular PCA method are evaluated under the conditions of varying expression, illumination and pose using standard face databases.
Keywords: PCA; Face recognition; Modular PCA; Pose invariance; Illumination invariance 1. Introduction
Face recognition is a difficult problem because of the generally similar shape of faces combined with the numerous variations between images of the same face. The image of a face changes with facial expression, age, viewpoint, illumination conditions, noise etc. The task of a face recognition system is to recognize a face in a manner that is as independent as possible of these image variations. Automatic recognition of faces is considered as one of the fundamental problems in computer vision and pattern analysis, and many scientists from different areas have addressed it. Chellappa et al. (1995) presented a survey on several statistical-based, neural network-based and feature-based methods for face recognition. Currently, one of the methods that yields promising results on frontal face recognition is the principal component analysis (PCA),which is a statistical approach where face images are expressed as a subset of their eigenvectors, and hence called eigenfaces (Sirovich and Kirby,1987; Turk and Pentland,1991;Moghaddam and Pentland,1997; Martinez, 2000; Graham and Allinson,1998).PCA has also been used for handprint recognition (Murase et al.,1981), human-made object recognition (Murase and Nayar, 1995), industrial robotics (Nayar et al.,1996), and mobile
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重庆邮电大学本科毕业设计（论文）
robotics (Weng,1996).But results show that the recognition rate is not satisfactory for pose variations exceeding 30° and extreme changes in illumination. The main objective of this research is to improve the accuracy of face recognition subjected to varying facial expression, illumination and head pose. As stated before, PCA method has been a popular technique in facial image recognition.
But this technique is not highly accurate when the illumination and pose of the facial images vary considerably. In this research work an attempt is made to improve the accuracy of this technique under the conditions of varying facial expression, illumination and pose. We propose the modular PCA method, which is an extension of the conventional PCA method. In the modular PCA method the face images are divided into smaller images and the PCA method is applied on each of them. Whereas in the traditional PCA method the entire face image is considered, hence large variation in pose or illumination will affect the recognition rate profoundly. Since in the case of modular PCA method the original face image is divided into sub-images the variations in pose or illumination in the image will affect only some of the sub- images, hence we expect this method to have better recognition rate than the conventional PCA method. A similar method called modular eigenspaces was proposed by Pentland et al. (1994).In this method PCA is performed on the eyes and nose of the face image.
This paper is organized as follows: Section 2 describes the conventional PCA method. Section 3 explains the modular PCA method. Section 4 describes the face databases used for testing the face recognition methods. Section 5 presents simulation results obtained by applying the PCA method and the proposed modular PCA method to the face image sets with large light and pose variations. Finally, a conclusion is drawn in Section 6. 2. Review of the PCA method
The PCA method has been extensively applied for the task of face recognition. Approximate reconstruction of faces in the ensemble was per- formed using a weighted combination of eigenvectors (eigenpictures), obtained from that ensemble (Sirovich and Kirby, 1987).The weights that characterize the expansion of the given image in terms of eigenpictures are seen as global facial features. In an extension of that work, Kirby and Sirovich (1990) included the inherent symmetry of faces in the eigenpictures.
All the face images in the face database are represented as very long vectors, instead of the usual matrix representation. This makes up the entire image space where each image is a point Since the faces have a similar structure (eye, nose and mouth, position, etc.), the vectors representing them will be correlated. We will see that faces of the same class will group at a certain location in the image space. Hence the face images are rep resented by a set of eigenvectors developed from a covariance matrix formed by the training of face images. The idea behind eigenimages (in our case eigenfaces) is to find a lower dimensional space in which shorter vectors will describe face images.Fig.1 illustrates this idea graphically.
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重庆邮电大学本科毕业设计（论文）
2.1. Computing eigenfaces
Consider the face images in the face database to be of size L by L. These images can be represented as a vector of dimension L2 ,or a point in L2–dimensional space. A set of images therefore corresponds to a set of points in this high dimensional space. Since facial images are similar in structure, these points will not be randomly distributed, and therefore can be described by a lower dimensional subspace.PCA gives the basis vectors for this subspace (which is called the ??face space??).Each basis vector is of length L2 , and is the eigenvector of the covariance matrix corresponding to the original face images.
Let I1,I2,...,IMbe the training set of face images. The average face is defined by
1A?M?Ii?1Mi （1）
Each face differs from the average face by the vectorYi?Ii?A.The covariance matrix C is obtained as
1C?M?Y.Yii?1MTi （2）
'The eigenvectors of the covariance matrix are computed and the Msignificant eigenvectors are
chosen as those with the largest corresponding eigenvalus. From these eigenvectors, the weights for each image in the training set are computed as
T WiK?EK.(Ii?A) ?i,K （3）
Where EK?s are the eigenvectors corresponding to theM'largest eigenvalues of C and K varies from 1 toM'. 2.2 Classi?cation
A test image Itest is projected into face space by the following operation:
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重庆邮电大学本科毕业设计（论文）
TWtestK?EK.(Itest?A) ?K （4）
p The weights WiK form a vectorTpT?[w1,w2,...,wM], which describes the contribution of each
'eigenface in representing the input face image. This vector can then be used to fit the test image to a predefined face class. A simple technique is to compute distance of WtestK fromTp, where Tp is the mean weight vector of the pth class. The test image can be classified to be in class p whenmin(Dp)??i, where Dp??Wtest?Tp?and ?iis the threshold. 3. Modular PCA method
The PCA based face recognition method is not very effective under the conditions of varying pose
and illumination, since it considers the global information of each face image and represents them with a set of weights. Under these conditions the weight vectors will vary considerably from the weight vectors of the images with normal pose and illumination, hence it is difficult to identify them correctly. On the other hand if the face images were divided into smaller regions and the weight vectors are computed for each of these regions, then the weights will be more representative of the local information of the face. When there is a variation in the pose or illumination, only some of the face regions will vary and rest of the regions will remain the same as the face regions of a normal image. Hence weights of the face regions not affected by varying pose and illumination will closely match with the weights of the same individual's face regions under normal conditions.Therefore it is expected that improved recognition rates can be obtained by following the modular PCA approach. We expect that if the face images are divided into very small regions the global information of the face may be lost and the accuracy of this method may deteriorate.
In this method, each image in the training set is divided into N smaller images. Hence the size of each sub-image will be L2 =N .These sub-images can be represented mathematically as
Iij(m,n)?Ii(LL(j?1)?m,(j?1)?n) ?i,j （5） NNwhere i varies from 1 to M, M being the number of images in the training set, j varies from 1 to N , N being the number of sub-images and m and n vary from 1 to Ldividing a face image into four smaller images using Eq.(5) for N=4. The average image of all the training sub-images is computed as
N.Fig2 shows the result of
1MN A???Iij （6）M.Ni?1j?1- 44 -
重庆邮电大学本科毕业设计（论文）
The next step is to normalize each training sub-image by subtracting it from the mean as
Yij?Iij?A ?i,j （7）
From the normalized sub-images the covariance matrix is computed as
1MN C?YijYijT （8）??M.Ni?1j?1'Next we find the eigenvectors of C that are associated with the Mlargest eigenvalues. We
represent the eigenvectors as E1,E2,...,EM.The weights are computed from the eigenvectors as
'shown below:
T WpnjK?EK.(Ipnj?A) ?p,n,j,K （9）
where K takes the values 1,2,…,M', n varies from 1 to?,?being the number of images per
individual, and p varies from 1 to P ,P being the number of individuals in the training set. Weights are also computed for the test sub-images using the eigenvectors as shown in the next equation:
TWtestjK?EK.(Itestj?A) ?j,K （10）
Mean weight set of each class in the training set is computed from the weight sets of the class as
shown below:
TpjK1M? ???WpnjK ?p,j （11）
?K?1n?1'Next the minimum distance is computed as shown below:
1MDpj?'?WtestjK?Tpjk （12）
MK?11N Dp??Dpj （13）
Nj?1min(Dp)??ifor a particular value of p, the corresponding face class in the training set is the
'closest one to the test image. Hence the test image is recognized as belonging to the pth face class. 4. Image databases
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