Extracting Discriminant Features of Faces Using Kernel Discriminant Analysis

Yongmin Li , Shaogang Gong and Heather Liddell
  1. Representation: PCA, LDA, KPCA, or KDA?
  2. A Toy Problem
  3. Extract the Discriminant Features of Multi-View Faces
  4. Relavant Publications

Representation: PCA, LDA, KPCA, or KDA?

Principal Component Analysis (PCA) has been widely adopted to reduce dimensionality and extract abstract features of faces. But the features extracted by PCA are actually ``global'' features for all face classes, thus they are not necessarily much representative for discriminating one face class from others. Linear Discriminant Analysis (LDA), which seeks to find a linear transformation by maximising the between-class variance and minimising the within-class variance, proved to be a suitable technique for face recognition. However, both the PCA and LDA are linear techniques which may be less efficient when severe non-linearity is involved. To extract the non-linear principal components, the Kernel PCA (KPCA) was developed for pattern recognition and has been adopted to construct a nonlinear models aiming at corresponding dynamic appearances of both shape and texture across views. However, similar to the linear PCA, KPCA captures the overall variance of all patterns which are not necessary significant for discriminant purpose. In this work, the Kernel Discriminant Analysis (KDA), a nonlinear discriminant approach based on the kernel technique which has been successfully used in KPCA is developed for extracting the nonlinear discriminant features

A Toy Problem

We use a toy problem to illustrate the characteristics of KDA as shown in Figure 1. Two classes of patterns denoted by circles and crosses respectively have a significant non-linear distribution. From left to right are the discriminant curves and the distribution of the one dimension feature of PCA, LDA, KPCA and KDA respectively. Among them, the KDA achieves the best performance: the discriminant curve accurately separates the two classes of patterns, and the feature intensity correctly reflects the actual pattern distribution.

Figure 1: Solving a nonlinear classification problem with PCA, LDA, KPCA and KDA.
\fbox{\includegraphics[width=.20\columnwidth]{figures/boundary_pca.eps}} \fbox{\includegraphics[width=.20\columnwidth]{figures/boundary_lda.eps}} \fbox{\includegraphics[width=.20\columnwidth]{figures/boundary_kpca.eps}} \fbox{\includegraphics[width=.20\columnwidth]{figures/boundary_kda.eps}}
\includegraphics[width=.22\columnwidth]{figures/distrib_pca.eps} \includegraphics[width=.22\columnwidth]{figures/distrib_lda.eps} \includegraphics[width=.22\columnwidth]{figures/distrib_kpca.eps} \includegraphics[width=.22\columnwidth]{figures/distrib_kda.eps}

Extract the Discriminant Features of Multi-View Faces

Modelling the appearance of faces across multiple views is much more challenging than that from a fixed view for the following reasons:
  1. the severe non-linearity caused by rotation in depth, self-occlusion, self-shading and illumination change;
  2. the appearance of different people from a same view is more similar than that of one person from different views.
In this work, we apply the KDA to extract the discriminant features for multi-view face recognition. The patterns used for face recognition is represented by the shape-and-pose-free texture patterns, which are extracted by fitting a multi-view dynamic face model on face images and warping them to the model mean shape in frontal view. Figure 2 shows the original face images, fitted multi-view face model overlaid on the face images, and the warped shape-and-pose-free texture patterns.

Figure 2: Extract the shape-and-pose-free texture patterns of multi-view face images using a multi-view face model.
\includegraphics[width=.10\textwidth]{figures/orig00.eps} \includegraphics[width=.10\textwidth]{figures/orig02.eps} \includegraphics[width=.10\textwidth]{figures/orig04.eps} \includegraphics[width=.10\textwidth]{figures/orig06.eps} \includegraphics[width=.10\textwidth]{figures/orig08.eps}

\includegraphics[width=.10\textwidth]{figures/mesh00.eps} \includegraphics[width=.10\textwidth]{figures/mesh02.eps} \includegraphics[width=.10\textwidth]{figures/mesh04.eps} \includegraphics[width=.10\textwidth]{figures/mesh06.eps} \includegraphics[width=.10\textwidth]{figures/mesh08.eps}

\includegraphics[width=.10\textwidth]{figures/warp00.eps} \includegraphics[width=.10\textwidth]{figures/warp02.eps} \includegraphics[width=.10\textwidth]{figures/warp04.eps} \includegraphics[width=.10\textwidth]{figures/warp06.eps} \includegraphics[width=.10\textwidth]{figures/warp08.eps}

We first apply the PCA to a set of these shape-and-pose-free texture patterns. Figure 3(a) illustrates the variation of the first PCA dimension with respect to the pose change. The patterns belonging to a same face class are linked together. Figure 3(b) shows the distribution of the texture patterns in the first two PCA dimension. It is noted that the variation from different face classes is not efficiently isolated from that from pose change, or more precisely, the former is even overwhelmed by the latter.

Figure 3: Multi-view face recognition problem: variation from different face classes vs. variation from pose change. (a) variation of the 1st PCA dimension wrt pose change. (b) pattern distribution in the first two PCA dimensions.
\includegraphics[width=.45\textwidth]{figures/pca_scatter1.eps} \includegraphics[width=.46\textwidth]{figures/pca_scatter2.eps}

Then we apply the KDA to the face patterns of the same face classes as shown in Figure 3. The variation and distribution of the patterns are shown in Figure 4(a) and Figure 4(b) respectively. Compared to the results of the PCA patterns in Figure 3, the improvement in terms of discriminant capability is significant. It is interesting to note from Figure 4 that the patterns of different face classes are separable when two KDA dimensions are used only, while these patterns are mingled together when the PCA is employed in Figure 3.

Figure 4: Distribution of the KDA patterns obtained from the same face images as in Figure 3.
\includegraphics[width=.45\textwidth]{figures/kda_dist1.eps} \includegraphics[width=.45\textwidth]{figures/kda_dist2.eps}

Relavant Publications

  1. Y. Li, S. Gong, and H. Liddell.
    Learning to recognise faces across views and over time using Kernel Discriminant Analysis.
    Technical report, Queen Mary, University of London, 2001.
  2. Y. Li, S. Gong, and H. Liddell.
    Modelling faces dynamically across views and over time.
    Technical report, Queen Mary, University of London, 2001.
Yongmin Li 2001-02-07