Kernel Discriminant Embedding in face recognition

Abstract

In this paper, we present a novel and effective feature extraction technique for face recognition. The proposed technique incorporates a kernel trick with Graph Embedding and the Fisher’s criterion which we call it as Kernel Discriminant Embedding (KDE). The proposed technique projects the original face samples onto a low dimensional subspace such that the within-class face samples are minimized and the between-class face samples are maximized based on Fisher’s criterion. The implementation of kernel trick and Graph Embedding criterion on the proposed technique reveals the underlying structure of data. Our experimental results on face recognition using ORL, FRGC and FERET databases validate the effectiveness of KDE for face feature extraction.

Introduction

Face recognition has received extensive attention from the public because of its wide applications in our daily life such as identity verification, human–computer interaction, surveillance, etc. The most popular technique in face recognition is the subspace based approach. This subspace approach stems from Turk and Pentland’s pioneering work which is known as Eigenface [1]. Essentially, Eigenface seeks an optimal linear transformation which aims to preserve the maximum variance of data. Fisherface (or Linear/Fisher Discriminant Analysis, L/FDA) is an improved version of Eigenface with incorporation of Fisher’s criterion. Fisherface method seeks a projection direction which maximizes the between-class scatter and minimizes the within-class scatter [2]. Typically, both techniques encode data using second order dependencies, i.e. pixel-wise scatter of the image data. Hence, they are insensitive to higher order dependencies which encode nonlinear relations among the pixels [3]. Therefore, Eigenface and Fisherface could be suboptimal for face recognition. Moreover, these techniques assume that data is inherently Gaussian which is usually not assured in practice.

Recent studies assume that high dimensional data is a set of geometrically associated points lying on or nearly on a low dimensional manifold [4], [5], [6]. Hence, the intrinsic geometrical structures of data may possess inherited discriminating power for classification. Some example algorithms, which reveal this intrinsic manifold structure of data, include the Laplacianface (or Locality Preserving Projection, LPP) [7] and the Neighbourhood Preserving Embedding (NPE) [8]. LPP preserves the neighbourhood structure of data set optimally based on a heat kernel nearest neighbour graph [7]. NPE takes into account the restriction that neighbouring points in the high dimensional space must remain within the same neighbourhood in the low dimension space and be located in a similar relative spatial situation (without changing the local structure of the nearest neighbours of each data point). Even though these methods are differentiated by their objective functions, they can be unified within Graph Embedding (GE) framework [9]. GE characterizes data points based on preserving properties of small neighbourhoods around the data points. The local neighbourhood relations illustrate the manifold embedding without any specific assumption on data distribution.

The underlying discriminating power of these algorithms may not always be assured since the real world data could be too complicated. In order to enhance the discriminating power of GE algorithms, a discriminant criterion has been explicitly integrated into the learning algorithms. For examples, Marginal Fisher Analysis (MFA) [9], Locality Sensitive Discriminant Analysis (LSDA) [10] and Neighbourhood Preserving Discriminant Embedding (NPDE) [11] incorporate the Fisher’s criterion (FC) for manifold learning. The Maximum Neighbourhood Margin Criterion (MNMC) as shown in [12] performs a manifold learning based on Maximum Margin Criterion (MMC). The promising discriminating capacity of these algorithms has been validated from the experimental results [9], [12]. However, these methods remained encoding pattern information based on second order dependencies; whereas those higher order dependencies in an image, such as the correlations among three or more pixels of an edge, which might capture pertinent information for recognition have been neglected [3]. Hence, a nonlinear mapping could be used to map the data in the input space to a higher dimensional feature space. The main purpose of mapping is to “unfold” the data manifold, so that those discriminative nonlinear structures can emerge under this new representation. The kernel trick allows this unfolding implicitly [3].