Face Recognition Using Laplacian faces

Abstract: The face recognition is a fairly controversial subject right now. A system such as this can recognize and track dangerous criminals and terrorists in a crowd, but some contend that it is an extreme invasion of privacy. The proponents of large-scale face recognition feel that it is a necessary evil to make our country safer. It could benefit the visually impaired and allow them to interact more easily with the environment. Also, a computer vision-based authentication system could be put in place to allow computer access or access to a specific room using face recognition. Another possible application would be to integrate this technology into an artificial intelligence system for more realistic interaction with humans.

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 Laplacian faces are the optimal linear approximations to the eigen functions 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 data sets. Experimental results suggest that the proposed Laplacianface approach provides a better representation and achieves lower error rates in face recognition. Principal Component Analysis (PCA) is a statistical method under the broad title of factor analysis. The purpose of PCA is to reduce the large dimensionality of the data space (observed variables) to the smaller intrinsic dimensionality of feature space (independent variables), which are needed to describe the data economically. This is the case when there is a strong correlation between observed variables. The jobs which PCA can do are prediction, redundancy removal, feature extraction, data compression, etc. Because PCA is a known powerful technique which can do something in the linear domain, applications having linear models are suitable, such as signal processing, image processing, system and control theory, communications, etc.

The main idea of using PCA for face recognition is to express the large 1-D vector of pixels constructed from 2-D face image into the compact principal components of the feature space. This is called eigenspace projection. Eigenspace is calculated by identifying the eigenvectors of the covariance matrix derived from a set of fingerprint images (vectors).








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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
1. 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.

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