24-01-2013, 04:30 PM
Face Recognition Using Laplacianfaces
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ABSTRACT
We propose an appearance based face recognition method called the Laplacianface approach. By using Locality Preserving
Projections (LPP), the face images are mapped into a face subspace for analysis. Different from Principal Component
Analysis (PCA) and Linear Discriminant Analysis (LDA) which effectively see only the Euclidean structure of face space,
LPP finds an embedding that preserves local information, and obtains a face subspace that best detects the essential face
manifold structure. The Laplacianfaces are the optimal linear approximations to the eigenfunctions of the Laplace Beltrami
operator on the face manifold. In this way, the unwanted variations resulting from changes in lighting, facial expression, and
pose may be eliminated or reduced. Theoretical analysis shows that PCA, LDA and LPP can be obtained from different
graph models. We compare the proposed Laplacianface approach with Eigenface and Fisherface methods on three different
face datasets. Experimental results suggest that the proposed Laplacianface approach provides a better representation and
achieves lower error rates in face recognition.
KEYWORDS
Face Recognition, Principal Component Analysis, Linear Discriminant Analysis, Locality Preserving
Projections, Face Manifold, Subspace Learning
INTRODUCTION
Many face recognition techniques have been developed over the past few decades. One of the
most successful and well-studied techniques to face recognition is the appearance-based method
[28][16]. When using appearance-based methods, we usually represent an image of size n×m pixels by a
vector in an n×m dimensional space. In practice, however, these n×m dimensional spaces are too large to
allow robust and fast face recognition. A common way to attempt to resolve this problem is to use dimensionality
reduction techniques [1][2][8][11][12][14][22][26][28][32][35]. Two of the most popular
techniques for this purpose are Principal Component Analysis (PCA) [28] and Linear Discriminant
Analysis (LDA) [2].
PCA is an eigenvector method designed to model linear variation in high-dimensional data. PCA
performs dimensionality reduction by projecting the original n-dimensional data onto the k (<<n)-
dimensional linear subspace spanned by the leading eigenvectors of the data’s covariance matrix. Its
goal is to find a set of mutually orthogonal basis functions that capture the directions of maximum variance
in the data and for which the coefficients are pairwise decorrelated. For linearly embedded manifolds,
PCA is guaranteed to discover the dimensionality of the manifold and produces a compact
representation. Turk and Pentland [28] use Principal Component Analysis to describe face images in
terms of a set of basis functions, or “eigenfaces”.
LDA is a supervised learning algorithm. LDA searches for the project axes on which the data
points of different classes are far from each other while requiring data points of the same class to be
close to each other. Unlike PCA which encodes information in an orthogonal linear space, LDA encodes
discriminating information in a linear separable space using bases are not necessarily orthogonal. It is
generally believed that algorithms based on LDA are superior to those based on PCA. However, some
recent work [14] shows that, when the training dataset is small, PCA can outperform LDA, and also that
PCA is less sensitive to different training datasets.
Recently, a number of research efforts have shown that the face images possibly reside on a
nonlinear submanifold [7][10][18][19][21][23][27]. However, both PCA and LDA effectively see only
the Euclidean structure. They fail to discover the underlying structure, if the face images lie on a
nonlinear submanifold hidden in the image space. Some nonlinear techniques have been proposed to
discover the nonlinear structure of the manifold, e.g. Isomap [27], LLE [18][20], and Laplacian
Eigenmap [3]. These nonlinear methods do yield impressive results on some benchmark artificial data
sets. However, they yield maps that are defined only on the training data points and how to evaluate the
maps on novel test data points remains unclear. Therefore, these nonlinear manifold learning techniques
[3][5][18][20] [27][33] might not be suitable for some computer vision tasks, such as face recognition.
In the meantime, there has been some interest in the problem of developing low dimensional representations
through kernel based techniques for face recognition [13][19]. These methods can discover
the nonlinear structure of the face images. However, they are computationally expensive. Moreover,
none of them explicitly considers the structure of the manifold on which the face images possibly reside.
In this paper, we propose a new approach to face analysis (representation and recognition), which
explicitly considers the manifold structure. To be specific, the manifold structure is modeled by a nearest-
neighbor graph which preserves the local structure of the image space. A face subspace is obtained
by Locality Preserving Projections (LPP) [9]. Each face image in the image space is mapped to a lowdimensional
face subspace, which is characterized by a set of feature images, called Laplacianfaces. The
face subspace preserves local structure, seems to have more discriminating power than the PCA approach
for classification purpose. We also provide theoretical analysis to show that PCA, LDA and LPP
can be obtained from different graph models. Central to this is a graph structure that is induced from the
data points. LPP finds a projection that respects this graph structure. In our theoretical analysis, we show
how PCA, LDA, and LPP arise from the same principle applied to different choices of this graph structure.
It is worthwhile to highlight several aspects of the proposed approach here:
1. While the Eigenfaces method aims to preserve the global structure of the image space, and the Fisherfaces
method aims to preserve the discriminating information; our Laplacianfaces method aims to
preserve the local structure of the image space. In many real world classification problems, the local
manifold structure is more important than the global Euclidean structure, especially when nearestneighbor
like classifiers are used for classification. LPP seems to have discriminating power although
it is unsupervised.
2. An efficient subspace learning algorithm for face recognition should be able to discover the nonlinear
manifold structure of the face space. Our proposed Laplacianfaces method explicitly considers the
manifold structure which is modeled by an adjacency graph. Moreover, the Laplacianfaces are obtained
by finding the optimal linear approximations to the eigenfunctions of the Laplace Beltrami operator
on the face manifold. They reflect the intrinsic face manifold structures.
3. LPP shares some similar properties to LLE, such as a locality preserving character. However, their
objective functions are totally different. LPP is obtained by finding the optimal linear approximations
to the eigenfunctions of the Laplace Beltrami operator on the manifold. LPP is linear, while LLE is
nonlinear. Moreover, LPP is defined everywhere, while LLE is defined only on the training data
points and it is unclear how to evaluate the maps for new test points. In contrast, LPP may be simply
applied to any new data point to locate it in the reduced representation space.
The rest of this paper is organized as follows: Section 2 describes PCA and LDA. The Locality
Preserving Projections (LPP) algorithm is described in Section 3. In Section 4, we provide a statistical
view of LPP. We then give a theoretical analysis of LPP and its connections to PCA and LDA in Section
5. Section 6 presents the manifold ways of face analysis using Laplacianfaces. A variety of experimental
results are presented in Section 7. Finally, we provide some concluding remarks and suggestions for future
work in Section 8.
PCA and LDA
One approach to coping with the problem of excessive dimensionality of the image space is to reduce
the dimensionality by combining features. Linear combinations are particular attractive because
they are simple to compute and analytically tractable. In effect, linear methods project the highdimensional
data onto a lower dimensional subspace.
Considering the problem of representing all of the vectors in a set of n d-dimensional samples x1,
x2, …, xn, with zero mean, by a single vector y={y1, y2, …, yn} such that yi represent xi. Specifically, we
find a linear mapping from the d-dimensional space to a line. Without loss of generality, we denote the
transformation vector by w. That is, wTxi = yi. Actually, the magnitude of w is of no real significance,
because it merely scales yi. In face recognition, each vector xi denotes a face image.
Different objective functions will yield different algorithms with different properties. PCA aims to
extract a subspace in which the variance is maximized. Its objective function is as follows,